Algebra of setsFrom Wikipedia, the free encyclopedia

Contents

1 Algebra of sets 11.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The fundamental laws of set algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The principle of duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Some additional laws for unions and intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Some additional laws for complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 The algebra of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 The algebra of relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Algebraic expression 62.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 In roots of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Algebraic vs. other mathematical expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Algebraic function 93.1 Algebraic functions in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 The role of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

i

ii CONTENTS

4 Algebraic operation 144.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Arithmetic vs algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Properties of arithmetic and algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Complement (set theory) 165.1 Relative complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Absolute complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4 Complements in various programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 List of types of functions 226.1 Relative to set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Relative to an operator (c.q. a group or other structure) . . . . . . . . . . . . . . . . . . . . . . . . 226.3 Relative to a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 Relative to an ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.5 Relative to the real/complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.6 Ways of defining functions/Relation to Type Theory . . . . . . . . . . . . . . . . . . . . . . . . . 236.7 Relation to Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Measurable function 257.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Caveat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 Special measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 Properties of measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 Non-measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Measure (mathematics) 288.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.3.2 Measures of infinite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 308.3.3 Measures of infinite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 30

CONTENTS iii

8.4 Sigma-finite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Vitali set 369.1 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 Construction and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 1

Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, andcomplementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu-ating expressions, and performing calculations, involving these operations and relations.Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union,the meet operator being intersection, and the complement operator being set complement.

1.1 Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition andmultiplicationare associative and commutative, so are set union and intersection; just as the arithmetic relation less than or equalis reflexive, antisymmetric and transitive, so is the set relation of subset.It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations ofequality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory,and for a full rigorous axiomatic treatment see axiomatic set theory.

1.2 The fundamental laws of set algebra

The binary operations of set union ( ) and intersection ( ) satisfy many identities. Several of these identities orlaws have well established names.

Commutative laws:

A B = B A A B = B A

Associative laws:

(A B) C = A (B C) (A B) C = A (B C)

Distributive laws:

A (B C) = (A B) (A C) A (B C) = (A B) (A C)

The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking.Like addition and multiplication, the operations of union and intersection are commutative and associative, and inter-section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

1

https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Equality_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Commutativityhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Axiomhttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Binary_operationhttps://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Commutative_operationhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Distributivity

2 CHAPTER 1. ALGEBRA OF SETS

Two additional pairs of laws involve the special sets called the empty set and the universal set U ; together withthe complement operator (AC denotes the complement of A). The empty set has no members, and the universal sethas all possible members (in a particular context).

Identity laws:

A = A A U = A

Complement laws:

A AC = U A AC =

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, and U are the identity elements for union and intersection, respectively.Unlike addition and multiplication, union and intersection do not have inverse elements. However the complementlaws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.The preceding five pairs of lawsthe commutative, associative, distributive, identity and complement lawsencompassall of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.Note that if the complement laws are weakened to the rule (AC)C = A , then this is exactly the algebra of proposi-tional linear logic.

1.3 The principle of duality

See also: Duality (order theory)

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other byinterchanging and , and also and U.These are examples of an extremely important and powerful property of set algebra, namely, the principle of dualityfor sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions andintersections, interchanging U and and reversing inclusions is also true. A statement is said to be self-dual if it isequal to its own dual.

1.4 Some additional laws for unions and intersections

The following proposition states six more important laws of set algebra, involving unions and intersections.PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:

idempotent laws:

A A = A A A = A

domination laws:

A U = U A =

absorption laws:

A (A B) = A A (A B) = A

https://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Universal_sethttps://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Identity_elementhttps://en.wikipedia.org/wiki/Inverse_elementhttps://en.wikipedia.org/wiki/Unary_operationhttps://en.wikipedia.org/wiki/Linear_logichttps://en.wikipedia.org/wiki/Duality_(order_theory)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Idempotenthttps://en.wikipedia.org/wiki/Absorption_law

1.5. SOME ADDITIONAL LAWS FOR COMPLEMENTS 3

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws statedabove. As an illustration, a proof is given below for the idempotent law for union.Proof:

The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law forunion, namely the idempotent law for intersection.Proof:

Intersection can be expressed in terms of set difference :A B = A (AB)

1.5 Some additional laws for complements

The following proposition states five more important laws of set algebra, involving complements.PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgans laws:

(A B)C = AC BC

(A B)C = AC BC

double complement or Involution law:

(AC)C = A

complement laws for the universal set and the empty set:

C = U UC =

Notice that the double complement law is self-dual.The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies thecomplement laws. In other words, complementation is characterized by the complement laws.PROPOSITION 5: Let A and B be subsets of a universe U, then:

uniqueness of complements:

If A B = U , and A B = , then B = AC

1.6 The algebra of inclusion

The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is apartial order.PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:

A A

antisymmetry:

A B and B A if and only if A = B

transitivity:

https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/De_Morgan%2527s_lawshttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relation

4 CHAPTER 1. ALGEBRA OF SETS

If A B and B C , then A C

The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, andhence together with the distributive and complement laws above, show that it is a Boolean algebra.PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:

A S

existence of joins:

A A B If A C and B C , then A B C

existence of meets:

A B A If C A and C B , then C A B

The following proposition says that the statement A B is equivalent to various other statements involving unions,intersections and complements.PROPOSITION 8: For any two sets A and B, the following are equivalent:

A B A B = A A B = B AB = BC AC

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of setunion or set intersection, which means that the notion of set inclusion is axiomatically superfluous.

1.7 The algebra of relative complements

The following proposition lists several identities concerning relative complements and set-theoretic differences.PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

C \ (A B) = (C \A) (C \B) C \ (A B) = (C \A) (C \B) C \ (B \A) = (A C) (C \B) (B \A) C = (B C) \A = B (C \A) (B \A) C = (B C) \ (A \ C) A \A = \A = A \ = A B \A = AC B (B \A)C = A BC

U \A = AC

A \ U =

https://en.wikipedia.org/wiki/Power_sethttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Complement_(set_theory)

1.8. SEE ALSO 5

1.8 See also -algebra is an algebra of sets, completed to include countably infinite operations.

Axiomatic set theory

Field of sets

Naive set theory

Set (mathematics)

1.9 References Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. TheAlgebra of Sets, pp 1623

Courant, Richard, Herbert Robbins, Ian Stewart,What is mathematics?: An Elementary Approach to Ideas andMethods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. SUPPLEMENT TO CHAPTER IITHE ALGEBRA OF SETS

1.10 External links Operations on Sets at ProvenMath

https://en.wikipedia.org/wiki/%CE%A3-algebrahttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Field_of_setshttps://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Special:BookSources/0486638294http://books.google.com/books?id=3-nrPB7BQKMC&pg=PA16#v=onepage&q&f=falsehttp://books.google.com/books?id=3-nrPB7BQKMC&pg=PA16#v=onepage&q&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/9780195105193http://books.google.com/books?id=UfdossHPlkgC&pg=PA17-IA8&dq=%2522algebra+of+sets%2522&hl=en&ei=k8-RTdXoF4K2tgfM-p1v&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDYQ6AEwAg#v=onepage&q=%2522algebra%2520of%2520sets%2522&f=falsehttp://books.google.com/books?id=UfdossHPlkgC&pg=PA17-IA8&dq=%2522algebra+of+sets%2522&hl=en&ei=k8-RTdXoF4K2tgfM-p1v&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDYQ6AEwAg#v=onepage&q=%2522algebra%2520of%2520sets%2522&f=falsehttp://www.apronus.com/provenmath/btheorems.htm

Chapter 2

Algebraic expression

Rational expression redirects here. For the notion in formal languages, see regular expression.

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraicoperations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational num-ber).[1] For example, 3x2 2xy+ c is an algebraic expression. Since taking the square root is the same as raising tothe power 12 ,

1 x21 + x2

is also an algebraic expression. By contrast, transcendental numbers like and e are not algebraic.A rational expression is an expression that may be rewritten to a rational fraction by using the properties of thearithmetic operations (commutative properties and associative properties of addition and multiplication, distributiveproperty and rules for the operations on the fractions). In other words, a rational expression is an expression whichmaybe constructed from the variables and the constants by using only the four operations of arithmetic. Thus, 3x

22xy+cy31

is a rational expression, whereas

1x21+x2 is not.

A rational equation is an equation in which two rational fractions (or rational expressions) of the form P (x)Q(x) areset equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.

2.1 Terminology

Algebra has its own terminology to describe parts of an expression:

1 Exponent (power), 2 coefficient, 3 term, 4 operator, 5 constant, x, y - variables

6

https://en.wikipedia.org/wiki/Regular_expressionhttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Expression_(mathematics)https://en.wikipedia.org/wiki/Constant_(mathematics)https://en.wikipedia.org/wiki/Variable_(mathematics)https://en.wikipedia.org/wiki/Algebraic_operationhttps://en.wikipedia.org/wiki/Algebraic_operationhttps://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Subtractionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Division_(mathematics)https://en.wikipedia.org/wiki/Exponentiationhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Square_roothttps://en.wikipedia.org/wiki/Transcendental_numberhttps://en.wikipedia.org/wiki/Pihttps://en.wikipedia.org/wiki/E_(mathematical_constant)https://en.wikipedia.org/wiki/Expression_(mathematics)https://en.wikipedia.org/wiki/Rational_fractionhttps://en.wikipedia.org/wiki/Commutative_propertyhttps://en.wikipedia.org/wiki/Associative_propertyhttps://en.wikipedia.org/wiki/Distributive_propertyhttps://en.wikipedia.org/wiki/Distributive_propertyhttps://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Rational_fractionhttps://en.wikipedia.org/wiki/Fraction_(mathematics)https://en.wikipedia.org/wiki/Cross-multiplicationhttps://en.wikipedia.org/wiki/Cross-multiplicationhttps://en.wikipedia.org/wiki/Algebra

2.2. IN ROOTS OF POLYNOMIALS 7

2.2 In roots of polynomials

The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can alwaysbe written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such asolution of an equation is called an algebraic solution. But the Abel-Ruffini theorem states that algebraic solutions donot exist for all such equations (just for some of them) if n 5.

2.3 Conventions

2.3.1 Variables

By convention, letters at the beginning of the alphabet (e.g. a, b, c ) are typically used to represent constants, andthose toward the end of the alphabet (e.g. x, y and z ) are used to represent variables.[2] They are usually written initalics.[3]

2.3.2 Exponents

By convention, terms with the highest power (exponent), are written on the left, for example, x2 is written to the leftof x . When a coefficient is one, it is usually omitted (e.g. 1x2 is written x2 ).[4] Likewise when the exponent (power)is one, (e.g. 3x1 is written 3x ),[5] and, when the exponent is zero, the result is always 1 (e.g. 3x0 is written 3 , sincex0 is always 1 ).[6]

2.4 Algebraic vs. other mathematical expressions

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions.A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient ofpolynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as x + 4.

2.5 See also Algebraic equation

Linear_equation#Algebraic_equations

Algebraic function

Analytical expression

Arithmetic expression

Closed-form expression

Expression (mathematics)

Polynomial

Term (logic)

2.6 Notes[1] Morris, Christopher G. (1992). Academic Press dictionary of science and technology. p. 74.

[2] William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, TheRosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71

https://en.wikipedia.org/wiki/Root_of_a_functionhttps://en.wikipedia.org/wiki/Degree_of_a_polynomialhttps://en.wikipedia.org/wiki/Polynomial_equationhttps://en.wikipedia.org/wiki/Quadratic_formulahttps://en.wikipedia.org/wiki/Cubic_functionhttps://en.wikipedia.org/wiki/Quartic_equationhttps://en.wikipedia.org/wiki/Algebraic_solutionhttps://en.wikipedia.org/wiki/Abel-Ruffini_theoremhttps://en.wikipedia.org/wiki/Mathematical_constanthttps://en.wikipedia.org/wiki/Variable_(mathematics)https://en.wikipedia.org/wiki/Exponentiationhttps://en.wikipedia.org/wiki/Quotienthttps://en.wikipedia.org/wiki/Polynomialhttps://en.wikipedia.org/wiki/Algebraic_equationhttps://en.wikipedia.org/wiki/Linear_equation#Algebraic_equationshttps://en.wikipedia.org/wiki/Algebraic_functionhttps://en.wikipedia.org/wiki/Analytical_expressionhttps://en.wikipedia.org/wiki/Arithmetic_expressionhttps://en.wikipedia.org/wiki/Closed-form_expressionhttps://en.wikipedia.org/wiki/Expression_(mathematics)https://en.wikipedia.org/wiki/Polynomialhttps://en.wikipedia.org/wiki/Term_(logic)http://books.google.co.uk/books?id=nauWlPTBcjIC&lpg=PA74&dq=algebraic%2520expression%2520over%2520a%2520field&pg=PA74#v=onepage&q&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/1615302190http://books.google.co.uk/books?id=ad0P0elU1_0C&lpg=PA71&dq=elementary%2520algebra%2520letters%2520alphabet%2520constants%2520variables&pg=PA71#v=onepage&q=letters&f=false

8 CHAPTER 2. ALGEBRAIC EXPRESSION

[3] James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425,9780387985428, 221 pages, [James E. Gentle page 183]

[4] DavidAlanHerzog, Teach Yourself Visually Algebra, Publisher JohnWiley&Sons, 2008, ISBN0470185597, 9780470185599,304 pages, page 72

[5] JohnC. Peterson, TechnicalMathematicsWith Calculus, Publisher Cengage Learning, 2003, ISBN0766861899, 9780766861893,1613 pages, page 31

[6] JeromeE.Kaufmann, Karen L. Schwitters,Algebra for College Students, Publisher Cengage Learning, 2010, ISBN0538733543,9780538733540, 803 pages, page 222

2.7 References James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. p. 8.

2.8 External links Weisstein, Eric W., Algebraic Expression, MathWorld.

https://en.wikipedia.org/wiki/Special:BookSources/0387985425https://en.wikipedia.org/wiki/Special:BookSources/0470185597http://books.google.co.uk/books?id=Igs6t_clf0oC&lpg=PA72&ots=Excnhf1AgW&dq=algebra%2520coefficient%2520one&pg=PA72#v=onepage&q=coefficient%2520of%25201&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/0766861899http://books.google.co.uk/books?id=PGuSDjHvircC&lpg=PA31&ots=NKrtZZ1KDE&dq=%2522when%2520the%2520exponent%2520is%25201%2522&pg=PA32#v=onepage&q=%2522when%2520the%2520exponent%2520is%25201%2522&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/0538733543http://books.google.co.uk/books?id=-AHtC0IYMhYC&lpg=PP1&ots=kL8erjajyR&dq=algebra%2520exponents%2520zero%2520one&pg=PA222#v=onepage&q=exponents%2520&f=falsehttp://books.google.co.uk/books?id=UyIfgBIwLMQC&lpg=PA8&dq=algebraic%2520expression%2520over%2520a%2520field&pg=PA8#v=onepage&q&f=falsehttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/AlgebraicExpression.htmlhttps://en.wikipedia.org/wiki/MathWorld

Chapter 3

Algebraic function

This article is about algebraic functions in calculus, mathematical analysis, and abstract algebra. For functions inelementary algebra, see function (mathematics).

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quiteoften algebraic functions can be expressed using a finite number of terms, involving only the algebraic operationsaddition, subtraction, multiplication, division, and raising to a fractional power:

f(x) = 1/x, f(x) =x, f(x) =

1 + x3

x3/7 7x1/3

are typical examples.However, some algebraic functions cannot be expressed by such finite expressions (as proven by Galois and NielsAbel), as it is for example the case of the function defined by

f(x)5 + f(x)4 + x = 0

In more precise terms, an algebraic function of degree n in one variable x is a function y = f(x) that satisfies apolynomial equation

an(x)yn + an1(x)y

n1 + + a0(x) = 0

where the coefficients ai(x) are polynomial functions of x, with coefficients belonging to a set S. Quite often, S = Q ,and one then talks about function algebraic overQ ", and the evaluation at a given rational value of such an algebraicfunction gives an algebraic number.A functionwhich is not algebraic is called a transcendental function, as it is for example the case of exp(x), tan(x), ln(x),(x). A composition of transcendental functions can give an algebraic function: f(x) = cos(arcsin(x)) =

1 x2 .

As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but nfunctions, sometimes also called branches. Consider for example the equation of the unit circle: y2 + x2 = 1. Thisdetermines y, except only up to an overall sign; accordingly, it has two branches: y =

1 x2.

An algebraic function in m variables is similarly defined as a function y which solves a polynomial equation in m +1 variables:

p(y, x1, x2, . . . , xm) = 0.

It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is thenguaranteed by the implicit function theorem.Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field ofrational functions K(x1,...,xm).

9

https://en.wikipedia.org/wiki/Calculushttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Elementary_algebrahttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Polynomial#Historyhttps://en.wikipedia.org/wiki/Algebraic_operationshttps://en.wikipedia.org/wiki/Galoishttps://en.wikipedia.org/wiki/Niels_Abelhttps://en.wikipedia.org/wiki/Niels_Abelhttps://en.wikipedia.org/wiki/Polynomial_equationhttps://en.wikipedia.org/wiki/Polynomial_functionhttps://en.wikipedia.org/wiki/Algebraic_numberhttps://en.wikipedia.org/wiki/Transcendental_functionhttps://en.wikipedia.org/wiki/Branch_cuthttps://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/Up_tohttps://en.wikipedia.org/wiki/Irreducible_polynomialhttps://en.wikipedia.org/wiki/Implicit_function_theoremhttps://en.wikipedia.org/wiki/Field_(mathematics)https://en.wikipedia.org/wiki/Algebraic_closurehttps://en.wikipedia.org/wiki/Rational_function

10 CHAPTER 3. ALGEBRAIC FUNCTION

3.1 Algebraic functions in one variable

3.1.1 Introduction and overview

The informal definition of an algebraic function provides a number of clues about the properties of algebraic func-tions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can beformed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this issomething of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem ofGalois theory), algebraic functions need not be expressible by radicals.First, note that any polynomial function y = p(x) is an algebraic function, since it is simply the solution y to theequation

y p(x) = 0.

More generally, any rational function y = p(x)q(x) is algebraic, being the solution to

q(x)y p(x) = 0.

Moreover, the nth root of any polynomial y = np(x) is an algebraic function, solving the equation

yn p(x) = 0.

Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solutionto

an(x)yn + + a0(x) = 0,

for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of xand y and gathering terms,

bm(y)xm + bm1(y)x

m1 + + b0(y) = 0.

Writing x as a function of y gives the inverse function, also an algebraic function.However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one.The inverse is the algebraic function x = y . Another way to understand this, is that the set of branches of thepolynomial equation defining our algebraic function is the graph of an algebraic curve.

3.1.2 The role of complex numbers

From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. Firstof all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence anypolynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions notexceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values.Thus, problems to do with the domain of an algebraic function can safely be minimized.Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express thefunction in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (seecasus irreducibilis). For example, consider the algebraic function determined by the equation

y3 xy + 1 = 0.

https://en.wikipedia.org/wiki/Algebraic_operationshttps://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Division_(mathematics)https://en.wikipedia.org/wiki/Nth_roothttps://en.wikipedia.org/wiki/Casus_irreducibilishttps://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theoryhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theoryhttps://en.wikipedia.org/wiki/Polynomial_functionhttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Horizontal_line_testhttps://en.wikipedia.org/wiki/One-to-one_functionhttps://en.wikipedia.org/wiki/Algebraic_curvehttps://en.wikipedia.org/wiki/Fundamental_theorem_of_algebrahttps://en.wikipedia.org/wiki/Algebraically_closed_fieldhttps://en.wikipedia.org/wiki/Domain_(mathematics)https://en.wikipedia.org/wiki/Casus_irreducibilis

3.1. ALGEBRAIC FUNCTIONS IN ONE VARIABLE 11

A graph of three branches of the algebraic function y, where y3 xy + 1 = 0, over the domain 3/22/3 < x < 50.

Using the cubic formula, we get

y = 2x3

108 + 1281 12x3

+3

108 + 1281 12x3

6.

For x 334 , the square root is real and the cubic root is thus well defined, providing the unique real root. On theother hand, for x > 334 , the square root is not real, and one has to choose, for the square root, either non real-squareroot. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the twoterms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanyingimage.It may be proven that there is no way to express this function in terms nth roots using real numbers only, even thoughthe resulting function is real-valued on the domain of the graph shown.On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complexanalysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraicfunction is in fact an analytic function, at least in the multiple-valued sense.Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 C is such that thepolynomial p(x0,y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhoodof x0. Choose a system of n non-overlapping discs i containing each of these zeros. Then by the argument principle

1

2i

i

py(x0, y)

p(x0, y)dy = 1.

By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x,y) has only one root in i, given bythe residue theorem:

fi(x) =1

2i

i

ypy(x, y)

p(x, y)dy

which is an analytic function.

https://en.wikipedia.org/wiki/Cubic_formulahttps://en.wikipedia.org/wiki/Complex_analysishttps://en.wikipedia.org/wiki/Complex_analysishttps://en.wikipedia.org/wiki/Argument_principlehttps://en.wikipedia.org/wiki/Analytic_functionhttps://en.wikipedia.org/wiki/Residue_theorem

12 CHAPTER 3. ALGEBRAIC FUNCTION

3.1.3 Monodromy

Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x),provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smallerthan the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminantvanishes. Hence there are only finitely many such points c1, ..., cm.A close analysis of the properties of the function elements fi near the critical points can be used to show that themonodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the entire functionassociated to the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points.Note that, away from the critical points, we have

p(x, y) = an(x)(y f1(x))(y f2(x)) (y fn(x))

since the fi are by definition the distinct zeros of p. The monodromy group acts by permuting the factors, and thusforms the monodromy representation of the Galois group of p. (The monodromy action on the universal coveringspace is related but different notion in the theory of Riemann surfaces.)

3.2 History

The ideas surrounding algebraic functions go back at least as far as Ren Descartes. The first discussion of algebraicfunctions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in whichhe writes:

let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methodsof division and extraction of roots, reduce it into an infinite series ascending or descending according tothe dimensions of x, and then find the integral of each of the resulting terms.

3.3 See also Algebraic expression

Analytic function

Complex function

Elementary function

Function (mathematics)

Generalized function

List of special functions and eponyms

List of types of functions

Polynomial

Rational function

Special functions

Transcendental function

3.4 References Ahlfors, Lars (1979). Complex Analysis. McGraw Hill.

van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer.

https://en.wikipedia.org/wiki/Discriminanthttps://en.wikipedia.org/wiki/Monodromy_theoremhttps://en.wikipedia.org/wiki/Ramification_(mathematics)https://en.wikipedia.org/wiki/Riemann_spherehttps://en.wikipedia.org/wiki/Entire_functionhttps://en.wikipedia.org/wiki/Monodromy_grouphttps://en.wikipedia.org/wiki/Galois_grouphttps://en.wikipedia.org/wiki/Monodromy_actionhttps://en.wikipedia.org/wiki/Universal_covering_spacehttps://en.wikipedia.org/wiki/Universal_covering_spacehttps://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttps://en.wikipedia.org/wiki/Edward_Waringhttps://en.wikipedia.org/wiki/Algebraic_expressionhttps://en.wikipedia.org/wiki/Analytic_functionhttps://en.wikipedia.org/wiki/Complex_functionhttps://en.wikipedia.org/wiki/Elementary_functionhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Generalized_functionhttps://en.wikipedia.org/wiki/List_of_special_functions_and_eponymshttps://en.wikipedia.org/wiki/List_of_types_of_functionshttps://en.wikipedia.org/wiki/Polynomialhttps://en.wikipedia.org/wiki/Rational_functionhttps://en.wikipedia.org/wiki/Special_functionshttps://en.wikipedia.org/wiki/Transcendental_functionhttps://en.wikipedia.org/wiki/Lars_Ahlforshttps://en.wikipedia.org/wiki/Bartel_Leendert_van_der_Waerden

3.5. EXTERNAL LINKS 13

3.5 External links Definition of Algebraic function in the Encyclopedia of Math

Weisstein, Eric W., Algebraic Function, MathWorld.

Algebraic Function at PlanetMath.org.

Definition of Algebraic function in David J. Darling's Internet Encyclopedia of Science

http://www.encyclopediaofmath.org/index.php/Algebraic_functionhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/AlgebraicFunction.htmlhttps://en.wikipedia.org/wiki/MathWorldhttp://planetmath.org/AlgebraicFunctionhttps://en.wikipedia.org/wiki/PlanetMathhttp://www.daviddarling.info/encyclopedia/A/algebraic_function.htmlhttps://en.wikipedia.org/wiki/David_J._Darling

Chapter 4

Algebraic operation

Algebraic operations in the solution to the quadratic equation. The radical sign, denoting a square root, is equivalent toexponentiation to the power of . The sign represents the equation written with either a + and with a - sign.

In mathematics, an algebraic operation is any one of the operations addition, subtraction, multiplication, division,raising to an integer power, and taking roots (fractional power). Algebraic operations are performed on an algebraicvariable, term or expression,[1] and work in the same way as arithmetic operations.[2]

4.1 Notation

Multiplication symbols are usually omitted, and implied when there is no operator between two variables or terms,or when a coefficient is used. For example, 3 x2 is written as 3x2, and 2 x y is written as 2xy.[3] Sometimesmultiplication symbols are replaced with either a dot, or center-dot, so that x y is written as either x . y or x y.Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol,[4]and it must be explicitly used, for example, 3x is written as 3 * x.Rather than using the obelus symbol, , division is usual represented with a vinculum, a horizontal line, e.g. 3/x + 1.In plain text and programming languages a slash (also called a solidus) is used, e.g. 3 / (x + 1).Exponents are usually formatted using superscripts, e.g. x2. In plain text, and in the TeX mark-up language, the caretsymbol, ^, represents exponents, so x2 is written as x ^ 2.[5][6] In programming languages such as Ada,[7] Fortran,[8]Perl,[9] Python[10] and Ruby,[11] a double asterisk is used, so x2 is written as x ** 2.The plus-minus sign, , is used as a shorthand notation for two expressions written as one, representing one expressionwith a plus sign, the other with a minus sign. For example y = x 1 represents the two equations y = x + 1 and y = x 1. Sometimes it is used for denoting positive-or-negative term such as x.

14

https://en.wikipedia.org/wiki/Quadratic_equationhttps://en.wikipedia.org/wiki/Square_roothttps://en.wikipedia.org/wiki/Exponentiationhttps://en.wikipedia.org/wiki/Plus-minus_signhttps://en.wikipedia.org/wiki/Mathematichttps://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Subtractionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Division_(mathematics)https://en.wikipedia.org/wiki/Exponentiationhttps://en.wikipedia.org/wiki/Nth_roothttps://en.wikipedia.org/wiki/Algebraic_expressionhttps://en.wikipedia.org/wiki/Coefficienthttps://en.wikipedia.org/wiki/Plain_texthttps://en.wikipedia.org/wiki/Programming_languageshttps://en.wikipedia.org/wiki/Calculatorshttps://en.wikipedia.org/wiki/Obelushttps://en.wikipedia.org/wiki/Vinculum_(symbol)https://en.wikipedia.org/wiki/Slash_(punctuation)https://en.wikipedia.org/wiki/Plain_texthttps://en.wikipedia.org/wiki/TeXhttps://en.wikipedia.org/wiki/Carethttps://en.wikipedia.org/wiki/Ada_(programming_language)https://en.wikipedia.org/wiki/Fortranhttps://en.wikipedia.org/wiki/Perlhttps://en.wikipedia.org/wiki/Python_(programming_language)https://en.wikipedia.org/wiki/Ruby_(programming_language)https://en.wikipedia.org/wiki/Plus-minus_sign

4.2. ARITHMETIC VS ALGEBRAIC OPERATIONS 15

4.2 Arithmetic vs algebraic operations

Algebraic operations work in the same way as arithmetic operations, as can be seen in the table below.Note: the use of the letters a and b is arbitrary, and the examples would be equally valid if we had used x and y .

4.3 Properties of arithmetic and algebraic operations

4.4 References[1] William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations"

[2] Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison,Phinney, Blakeman, & Co., 1866, page 7

[3] Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, Algebraic notation, in Mathematics Matters Secondary 1 Express Text-book, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68

[4] William P. Berlinghoff, Fernando Q. Gouva,Math through the Ages: A Gentle History for Teachers and Others, PublisherMAA, 2004, ISBN 0883857367, 9780883857366, page 75

[5] Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153,9789380298153, page 212

[6] George Grtzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17

[7] S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual,Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352,page 13

[8] C. Xavier, Fortran 77AndNumericalMethods, PublisherNewAge International, 1994, ISBN812240670X, 9788122406702,page 20

[9] Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140,9781449313142, page 24

[10] MatthewA. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN1598631586,9781598631586, page 46

[11] Kevin C. Baird,Ruby by Example: Concepts and Code, PublisherNo Starch Press, 2007, ISBN1593271484, 9781593271480,page 72

[12] Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: CengageLearning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 7

4.5 See also Elementary algebra

Order of operations

https://en.wikipedia.org/wiki/Arithmetic_operation#Arithmetic_operationshttp://books.google.co.uk/books?id=BqQZAAAAYAAJ&lpg=PA55&ots=ex07zH_ljg&dq=%2522Algebraic%2520operations%2522&pg=PA55#v=onepage&q=%2522Algebraic%2520operations%2522&f=falsehttp://books.google.co.uk/books?id=dKZXAAAAYAAJ&dq=Elementary%2520algebra%2520notation&pg=PA7#v=onepage&q=Elementary%2520algebra%2520notation&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/9812738827http://books.google.co.uk/books?id=nL5ObMmDvPEC&lpg=PR9-IA8&ots=T_h6l40AE5&dq=%2522Algebraic%2520notation%2522%2520multiplication%2520omitted&pg=PR9-IA8#v=onepage&q=%2522Algebraic%2520notation%2522%2520multiplication%2520omitted&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/0883857367http://books.google.co.uk/books?id=JAXNVaPt7uQC&lpg=PA75&ots=-P78Lrz792&dq=calculator%2520asterisk%2520multiplication&pg=PA75#v=onepage&q=calculator%2520asterisk%2520multiplication&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/9380298153http://books.google.co.uk/books?id=zQa5I2sHPKEC&lpg=PA212&ots=s6pWav1Z_D&dq=%2522plain%2520text%2522%2520math%2520caret%2520exponent&pg=PA212#v=onepage&q=exponentiation%2520caret&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/0817641327http://books.google.co.uk/books?id=mLdg5ZdDKToC&lpg=PP1&ots=V9DFIaAAh0&dq=tex%2520math&pg=PA17#v=onepage&q=subscripts%2520and%2520superscripts%2520caret&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/3540693351http://books.google.co.uk/books?id=694P3YtXh-0C&lpg=PA718&ots=O_EgQ75FeB&dq=ada%2520%2520asterisk&pg=PA12#v=onepage&q=double%2520star%2520exponentiate&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/812240670Xhttp://books.google.co.uk/books?id=WYMgF9WFty0C&lpg=PA20&ots=BTtzs9F-NB&dq=fortran%2520asterisk%2520exponentiation&pg=PA20#v=onepage&q=fortran%2520asterisk%2520exponentiation&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/1449313140http://books.google.co.uk/books?id=l2IwEuRjeNwC&lpg=PA24&ots=5nsYOLHxlD&dq=perl%2520asterisk%2520exponentiation&pg=PA24#v=onepage&q=double%2520asterisk%2520exponentiation&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/1598631586http://books.google.co.uk/books?id=754knV_fyf8C&lpg=PA46&ots=8fEi1F-H8-&dq=python%2520asterisk%2520exponentiation&pg=PA46#v=onepage&q=double%2520asterisk%2520exponentiation&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/1593271484http://books.google.co.uk/books?id=kq2dBNdAl3IC&lpg=PA72&ots=0UU3k-Pvh8&dq=ruby%2520asterisk%2520exponentiation&pg=PA72#v=onepage&q=double%2520asterisk%2520exponentiation&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/061885195Xhttp://books.google.co.uk/books?id=5iXVZHhkjAgC&lpg=PA6&ots=iwrSrCrrOb&dq=operations%2520addition%252C%2520subtraction%252C%2520multiplication%252C%2520division%2520exponentiation.&pg=PA7#v=onepage&q=associative%2520property&f=falsehttps://en.wikipedia.org/wiki/Elementary_algebrahttps://en.wikipedia.org/wiki/Order_of_operations

Chapter 5

Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complementof A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are consideredto be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.

5.1 Relative complement

If A and B are sets, then the relative complement of A in B,[1] also termed the set-theoretic difference of B andA,[2] is the set of elements in B, but not in A.

The relative complement of A (left circle) in B (right circle): B Ac = B \A

The relative complement of A in B is denoted B A according to the ISO 31-11 standard (sometimes written B A,but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b a, where b is taken fromB and a from A).

16

https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/ISO_31-11#Sets

5.2. ABSOLUTE COMPLEMENT 17

Formally

B \A = {x B |x / A}.

Examples:

{1,2,3} {2,3,4} = {1} {2,3,4} {1,2,3} = {4} If R is the set of real numbers and Q is the set of rational numbers, then R \ Q = I is the set ofirrational numbers.

The following lists some notable properties of relative complements in relation to the set-theoretic operations of unionand intersection.If A, B, and C are sets, then the following identities hold:

C (A B) = (C A)(C B) C (A B) = (C A)(C B) C (B A) = (C A)(C B)

[ Alternately written: A (B C) = (A B)(A C) ]

(B A) C = (B C) A = B(C A) (B A) C = (B C) (A C) A A = A = A = A

5.2 Absolute complement

If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simplycomplement) of A, and is denoted by Ac or sometimes A. The same set often[3] is denoted by UA or A if U isfixed, that is:

Ac = U A.

For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of evennumbers.The following lists some important properties of absolute complements in relation to the set-theoretic operations ofunion and intersection.If A and B are subsets of a universe U, then the following identities hold:

De Morgans laws:[1]

(A B)c = Ac Bc. (A B)c = Ac Bc.

Complement laws:[1]

A Ac = U. A Ac = . c = U.

https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Irrational_numberhttps://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Universe_(mathematics)https://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Universe_(mathematics)https://en.wikipedia.org/wiki/De_Morgan%2527s_laws

18 CHAPTER 5. COMPLEMENT (SET THEORY)

The absolute complement of A in U : Ac = U \A

U c = . IfA B then ,Bc Ac.

(this follows from the equivalence of a conditional with its contrapositive)

Involution or double complement law:

(Ac)c = A.

Relationships between relative and absolute complements:

A B = A Bc (A B)c = Ac B

The first two complement laws above shows that if A is a non-empty, proper subset of U, then {A, Ac} is a partitionof U.

5.3 Notation

In the LaTeX typesetting language, the command \setminus[4] is usually used for rendering a set difference symbol,which is similar to a backslash symbol. When rendered the \setminus command looks identical to \backslash exceptthat it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. Avariant \smallsetminus is available in the amssymb package.

5.4 Complements in various programming languages

Some programming languages allow for manipulation of sets as data structures, using these operators or functions toconstruct the difference of sets a and b:

https://en.wikipedia.org/wiki/Contrapositivehttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Proper_subsethttps://en.wikipedia.org/wiki/Partition_of_a_sethttps://en.wikipedia.org/wiki/LaTeXhttps://en.wikipedia.org/wiki/Backslashhttps://en.wikipedia.org/wiki/Set_(computer_science)

5.4. COMPLEMENTS IN VARIOUS PROGRAMMING LANGUAGES 19

.NET Framework a.Except(b);

C++ set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());

Clojure (clojure.set/difference a b)[5]

Common Lisp set-difference, nset-difference[6]

F# Set.difference a b[7]

or

a - b[8]

Falcon diff = a - b[9]

Haskell difference a b

a \\ b[10]

Java diff = a.clone();

diff.removeAll(b);[11]

Julia setdiff[12]

Mathematica Complement[13]

MATLAB setdiff[14]

OCaml Set.S.diff[15]

Octave setdiff[16]

Pascal SetDifference := a - b;

Perl 5 #for perl version >= 5.10

@a = grep {not $_ ~~ @b} @a;

Perl 6 $A $B

$A (-) $B # texas version

PHP array_diff($a, $b);[17]

Prolog a(X),\+ b(X).

Python diff = a.difference(b)[18]

diff = a - b[18]

R setdiff[19]

Racket (set-subtract a b)[20]

https://en.wikipedia.org/wiki/.NET_Frameworkhttps://en.wikipedia.org/wiki/C++https://en.wikipedia.org/wiki/Clojurehttps://en.wikipedia.org/wiki/Common_Lisphttps://en.wikipedia.org/wiki/F#_(programming_language)https://en.wikipedia.org/wiki/Falcon_(programming_language)https://en.wikipedia.org/wiki/Haskell_(programming_language)https://en.wikipedia.org/wiki/Java_(programming_language)https://en.wikipedia.org/wiki/Julia_(programming_language)https://en.wikipedia.org/wiki/Mathematicahttps://en.wikipedia.org/wiki/MATLABhttps://en.wikipedia.org/wiki/OCamlhttps://en.wikipedia.org/wiki/GNU_Octavehttps://en.wikipedia.org/wiki/Pascal_(programming_language)https://en.wikipedia.org/wiki/Perl_5https://en.wikipedia.org/wiki/Perl_6https://en.wikipedia.org/wiki/PHPhttps://en.wikipedia.org/wiki/Prologhttps://en.wikipedia.org/wiki/Python_(programming_language)https://en.wikipedia.org/wiki/R_(programming_language)https://en.wikipedia.org/wiki/Racket_(programming_language)

20 CHAPTER 5. COMPLEMENT (SET THEORY)

Ruby diff = a - b[21]

Scala a.diff(b)[22]

or

a -- b[22]

Smalltalk (Pharo) a difference: b

SQL SELECT * FROM A

EXCEPT SELECT * FROM B

Unix shell comm 23 a b[23]

grep -vf b a # less efficient, but works with small unsorted sets

5.5 See also Algebra of sets

Naive set theory

Symmetric difference

5.6 References[1] Halmos (1960) p.17

[2] Devlin (1979) p.6

[3] Bourbaki p. E II.6

[4] The Comprehensive LaTeX Symbol List

[5] clojure.set API reference

[6] Common Lisp HyperSpec, Function set-difference, nset-difference. Accessed on September 8, 2009.

[7] Set.difference Method (F#). Accessed on July 12, 2015.

[9] Array subtraction, data structures. Accessed on July 28, 2014.

[10] Data.Set (Haskell)

[11] Set (Java 2 Platform SE 5.0). JavaTM 2 Platform Standard Edition 5.0 API Specification, updated in 2004. Accessed onFebruary 13, 2008.

[12] . The Standard Library--Julia Language documentation. Accessed on September 24, 2014

[13] Complement. Mathematica Documentation Center for version 6.0, updated in 2008. Accessed on March 7, 2008.

[14] Setdiff. MATLAB Function Reference for version 7.6, updated in 2008. Accessed on May 19, 2008.

[15] Set.S (OCaml).

[16] . GNU Octave Reference Manual

[17] PHP: array_diff, PHP Manual

https://en.wikipedia.org/wiki/Ruby_(programming_language)https://en.wikipedia.org/wiki/Scala_(programming_language)https://en.wikipedia.org/wiki/Smalltalk_(Pharo)https://en.wikipedia.org/wiki/SQLhttps://en.wikipedia.org/wiki/Unix_shellhttps://en.wikipedia.org/wiki/Algebra_of_setshttps://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Symmetric_differencehttp://www.lispworks.com/documentation/HyperSpec/Body/f_set_di.htmhttps://msdn.microsoft.com/en-us/library/ee340332.aspx,https://msdn.microsoft.com/en-us/library/ee353414.aspx,http://falconpl.org/index.ftd?page_id=sitewiki&prj_id=_falcon_site&sid=wiki&pwid=Survival%2520Guide&wid=Survival%253ABasic+Structures#Arrays,http://haskell.org/ghc/docs/latest/html/libraries/containers/Data-Set.htmlhttp://java.sun.com/j2se/1.5.0/docs/api/java/util/Set.htmlhttp://reference.wolfram.com/mathematica/ref/Complement.htmlhttp://www.mathworks.com/access/helpdesk/help/techdoc/ref/setdiff.htmlhttp://caml.inria.fr/pub/docs/manual-ocaml/libref/Set.S.htmlhttp://php.net/manual/en/function.array-diff.php

5.7. EXTERNAL LINKS 21

[18] . Python v2.7.3 documentation. Accessed on January 17, 2013.

[19] R Reference manual p. 410.

[20] . The Racket Reference. Accessed on May 19, 2015.

[21] Class: Array Ruby Documentation

[22] scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.

[23] comm(1), Unix Seventh Edition Manual, 1979.

Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van NostrandCompany. Zbl 0087.04403.

Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN0-387-90441-7. Zbl 0407.04003.

Bourbaki, N. (1970). Thorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.

5.7 External links Weisstein, Eric W., Complement, MathWorld.

Weisstein, Eric W., Complement Set, MathWorld.

http://cran.r-project.org/doc/manuals/fullrefman.pdfhttp://www.ruby-doc.org/core/classes/Array.htmlhttp://www.scala-lang.org/api/current/index.html#scala.collection.Set,http://plan9.bell-labs.com/7thEdMan/https://en.wikipedia.org/wiki/Paul_Halmoshttps://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0087.04403https://en.wikipedia.org/wiki/Keith_Devlinhttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-90441-7https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0407.04003https://en.wikipedia.org/wiki/Nicolas_Bourbakihttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-34034-8https://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/Complement.htmlhttps://en.wikipedia.org/wiki/MathWorldhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/ComplementSet.htmlhttps://en.wikipedia.org/wiki/MathWorld

Chapter 6

List of types of functions

Functions can be identified according to the properties they have. These properties describe the functions behaviourunder certain conditions. A parabola is a specific type of function.

6.1 Relative to set theory

These properties concern the domain, the codomain and the range of functions.

Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.In other words, every element of the functions codomain is the image of at most one elementof its domain.

Surjective function: has a preimage for every element of the codomain, i.e. the codomain equals the range.Also called a surjection or onto function.

Bijective function: is both an injection and a surjection, and thus invertible.

Identity function: maps any given element to itself.

Constant function: has a fixed value regardless of arguments.

Empty function: whose domain equals the empty set.

6.2 Relative to an operator (c.q. a group or other structure)

These properties concern how the function is affected by arithmetic operations on its operand.The following are special examples of a homomorphism on a binary operation:

Additive function: preserves the addition operation: f(x + y) = f(x) + f(y).

Multiplicative function: preserves the multiplication operation: f(xy) = f(x)f(y).

Relative to negation:

Even function: is symmetric with respect to the Y-axis. Formally, for each x: f(x) = f(x).

Odd function: is symmetric with respect to the origin. Formally, for each x: f(x) = f(x).

Relative to a binary operation and an order:

Subadditive function: for which the value of f(x+y) is less than or equal to f(x) + f(y).

Superadditive function: for which the value of f(x+y) is greater than or equal to f(x) + f(y).

22

https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Domain_(mathematics)https://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Range_(mathematics)https://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Surjective_functionhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Onto_functionhttps://en.wikipedia.org/wiki/Bijective_functionhttps://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Surjectionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Identity_functionhttps://en.wikipedia.org/wiki/Constant_functionhttps://en.wikipedia.org/wiki/Empty_functionhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Mathematical_structurehttps://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Homomorphismhttps://en.wikipedia.org/wiki/Binary_operationhttps://en.wikipedia.org/wiki/Additive_maphttps://en.wikipedia.org/wiki/Multiplicative_functionhttps://en.wikipedia.org/wiki/Negationhttps://en.wikipedia.org/wiki/Even_functionhttps://en.wikipedia.org/wiki/Odd_functionhttps://en.wikipedia.org/wiki/Origin_(mathematics)https://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Subadditive_functionhttps://en.wikipedia.org/wiki/Superadditive_function

6.3. RELATIVE TO A TOPOLOGY 23

6.3 Relative to a topology Continuous function: in which preimages of open sets are open.

Nowhere continuous function: is not continuous at any point of its domain (e.g. Dirichlet function).

Homeomorphism: is an injective function that is also continuous, whose inverse is continuous.

6.4 Relative to an ordering Monotonic function: does not reverse ordering of any pair.

Strict Monotonic function: preserves the given order.

6.5 Relative to the real/complex numbers Analytic function: Can be defined locally by a convergent power series.

Arithmetic function: A function from the positive integers into the complex numbers.

Differentiable function: Has a derivative.

Smooth function: Has derivatives of all orders.

Holomorphic function: Complex valued function of a complex variable which is differentiable at every pointin its domain.

Meromorphic function: Complex valued function that is holomorphic everywhere, apart from at isolated pointswhere there are poles.

Entire function: A holomorphic function whose domain is the entire complex plane.

6.6 Ways of defining functions/Relation to Type Theory Composite function: is formed by the composition of two functions f and g, by mapping x to f(g(x)).

Inverse function: is declared by doing the reverse of a given function (e.g. arcsine is the inverse of sine).

Piecewise function: is defined by different expressions at different intervals.

In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating whatit should map to. For this purpose, the 7 symbol or Church's is often used. Also, sometimes mathematiciansnotate a functions domain and codomain by writing e.g. f : A B . These notions extend directly to lambdacalculus and type theory, respectively.

6.7 Relation to Category Theory

Category Theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms.A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a setof morphisms. A partial (equiv. dependently typed) binary operation called composition is provided on morphisms,every object has one special morphism from it to itself called the identity on that object, and composition and identitiesare required to obey certain relations.In a so-called concrete category, the objects are associated with mathematical structures like sets, magmas, groups,rings, topological spaces, vector spaces, metric spaces, partial orders, differentiable manifolds, uniform spaces, etc.,and morphisms between two objects are associated with structure-preserving functions between them. In the ex-amples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms,

https://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Nowhere_continuoushttps://en.wikipedia.org/wiki/Dirichlet_functionhttps://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Analytic_functionhttps://en.wikipedia.org/wiki/Convergent_serieshttps://en.wikipedia.org/wiki/Power_serieshttps://en.wikipedia.org/wiki/Arithmetic_functionhttps://en.wikipedia.org/wiki/Integershttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Differentiable_functionhttps://en.wikipedia.org/wiki/Derivativehttps://en.wikipedia.org/wiki/Smooth_functionhttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Meromorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Pole_(complex_analysis)https://en.wikipedia.org/wiki/Entire_functionhttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Composite_functionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Arcsinehttps://en.wikipedia.org/wiki/Sinehttps://en.wikipedia.org/wiki/Piecewise_functionhttps://en.wikipedia.org/wiki/Alonzo_Churchhttps://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Lambda_calculushttps://en.wikipedia.org/wiki/Lambda_calculushttps://en.wikipedia.org/wiki/Type_theoryhttps://en.wikipedia.org/wiki/Category_Theoryhttps://en.wikipedia.org/wiki/Morphismshttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Dependently_typedhttps://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Concrete_categoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Magmashttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Vector_spaceshttps://en.wikipedia.org/wiki/Metric_spaceshttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Differentiable_manifoldshttps://en.wikipedia.org/wiki/Uniform_spaceshttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Homomorphismshttps://en.wikipedia.org/wiki/Group_homomorphisms

24 CHAPTER 6. LIST OF TYPES OF FUNCTIONS

continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable func-tions, and uniformly continuous functions, respectively.As an algebraic theory, one of the advantages of category theory is to enable one to prove many general resultswith a minimum of assumptions. Many common notions from mathematics (e.g. surjective, injective, free ob-ject, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism,epimorphism).Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf.topos).Allegory theory[1] provides a generalization comparable to category theory for relations instead of functions.

6.8 References[1] Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. ISBN 978-0-

444-70368-2.

https://en.wikipedia.org/wiki/Continuous_functionshttps://en.wikipedia.org/wiki/Linear_transformationshttps://en.wikipedia.org/wiki/Matrix_(mathematics)https://en.wikipedia.org/wiki/Metric_maphttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Differentiablehttps://en.wikipedia.org/wiki/Uniformly_continuoushttps://en.wikipedia.org/wiki/Surjectivehttps://en.wikipedia.org/wiki/Injectivehttps://en.wikipedia.org/wiki/Free_objecthttps://en.wikipedia.org/wiki/Free_objecthttps://en.wikipedia.org/wiki/Basis_(linear_algebra)https://en.wikipedia.org/wiki/Group_representationhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Monomorphismhttps://en.wikipedia.org/wiki/Epimorphismhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Type_theoryhttps://en.wikipedia.org/wiki/Toposhttps://en.wikipedia.org/wiki/Allegory_(category_theory)https://en.wikipedia.org/wiki/Relation_(mathematics)https://en.wikipedia.org/wiki/Special:BookSources/9780444703682https://en.wikipedia.org/wiki/Special:BookSources/9780444703682

Chapter 7

Measurable function

A function is Lebesgue measurable if and only if the preimage of each of the sets [a,] is a Lebesgue measurable set.

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions betweenmeasurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function betweenmeasurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to thesituation of continuous functions between topological spaces.In probability theory, the sigma algebra often represents the set of available information, and a function (in thiscontext a random variable) is measurable if and only if it represents an outcome that is knowable based on theavailable information. In contrast, functions that are not Lebesgue measurable are generally considered pathological,at least in the field of analysis.

7.1 Formal definition

Let (X, ) and (Y, ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras and . A function f: X Y is said to be measurable if the preimage of E under f is in for every E ; i.e.

f1(E) := {x X| f(x) E} , E T.

The notion of measurability depends on the sigma algebras and . To emphasize this dependency, if f: X Y isa measurable function, we will write

f : (X,) (Y, T )

25

https://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Lebesgue_measurable_sethttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Measure_theoryhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Measurable_spacehttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Measurable_sethttps://en.wikipedia.org/wiki/Measurablehttps://en.wikipedia.org/wiki/Continuity_(topology)https://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Pathological_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysis

26 CHAPTER 7. MEASURABLE FUNCTION

7.2 Caveat

This definition can be deceptively simple, however, as special care must be taken regarding the -algebras involved.In particular, when a function f: RR is said to be Lebesgue measurable what is actually meant is that f : (R,L) (R,B) is a measurable functionthat is, the domain and range represent different -algebras on the same underlyingset (here L is the sigma algebra of Lebesgue measurable sets, and B is the Borel algebra on R). As a result, thecomposition of Lebesgue-measurable functions need not be Lebesgue-measurable.By convention a topological space is assumed to be equipped with the Borel algebra generated by its open subsetsunless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to referonly to real-valued measurable functions with respect to the Borel algebra.[1] If the values of the function lie in aninfinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weakmeasurability and Bochner measurability.

7.3 Special measurable functions If (X, ) and (Y, ) are Borel spaces, a measurable function f: (X, ) (Y, ) is also called a Borel function.Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurablefunction is nearly a continuous function; see Luzins theorem. If a Borel function happens to be a section ofsome map Y X , it is called a Borel section.

ALebesgue measurable function is a measurable function f : (R,L) (C,BC) , whereL is the sigma algebraof Lebesgue measurable sets, and BC is the Borel algebra on the complex numbers C. Lebesgue measurablefunctions are of interest in mathematical analysis because they can be integrated. In the case f : X R, f is Lebesgue measurable iff {f > } = {x X : f(x) > } is measurable for all real . This isalso equivalent to any of {f }, {f < }, {f } being measurable for all . Continuous functions,monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functionsof bounded variation are all Lebesgue measurable.[2] A function f : X C is measurable iff the real andimaginary parts are measurable.

Random variables are by definition measurable functions defined on sample spaces.

7.4 Properties of measurable functions The sum and product of two complex-valued measurable functions are measurable.[3] So is the quotient, solong as there is no division by zero.[1]

The composition of measurable functions is measurable; i.e., if f: (X, 1) (Y, 2) and g: (Y, 2) (Z,3) are measurable functions, then so is g(f()): (X, 1) (Z, 3).[1] But see the caveat regarding Lebesgue-measurable functions in the introduction.

The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) ofreal-valued measurable functions are all measurable as well.[1][4]

The pointwise limit of a sequence of measurable functions is measurable (if the codomain in endowed withthe Borel algebra); note that the corresponding statement for continuous functions requires stronger conditionsthan pointwise convergence, such as uniform convergence. (This is correct when the counter domain of theelements of the sequence is a metric space. It is false in general; see pages 125 and 126 of.[5])

7.5 Non-measurable functions

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.

https://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Sigma_algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Real_numbershttps://en.wikipedia.org/wiki/Complex_numbershttps://en.wikipedia.org/wiki/Borel_sethttps://en.wikipedia.org/wiki/Infinite-dimensional_vector_spacehttps://en.wikipedia.org/wiki/Weak_measurabilityhttps://en.wikipedia.org/wiki/Weak_measurabilityhttps://en.wikipedia.org/wiki/Bochner_measurabilityhttps://en.wikipedia.org/wiki/Borel_spacehttps://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Luzin%2527s_theoremhttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Sigma_algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Lebesgue_integrationhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Sample_spacehttps://en.wikipedia.org/wiki/Supremumhttps://en.wikipedia.org/wiki/Infimumhttps://en.wikipedia.org/wiki/Limit_superiorhttps://en.wikipedia.org/wiki/Limit_inferiorhttps://en.wikipedia.org/wiki/Pointwise

7.6. SEE ALSO 27

So long as there are non-measurable sets in a measure space, there are non-measurable functions from thatspace. If (X, ) is some measurable space and A X is a non-measurable set, i.e. if A , then the indicatorfunction 1A: (X, ) R is non-measurable (where R is equipped with the Borel algebra as usual), since thepreimage of the measurable set {1} is the non-measurable set A. Here 1A is given by

1A(x) ={1 if x A0 otherwise

Any non-constant function can be made non-measurable by equipping the domain and range with appropriate-algebras. If f: X R is an arbitrary non-constant, real-valued function, then f is non-measurable if X isequipped with the indiscrete algebra = {, X}, since the preimage of any point in the range is some proper,nonempty subset of X, and therefore does not lie in .

7.6 See also Vector spaces of measurable functions: the Lp spaces

Measure-preserving dynamical system

7.7 Notes[1] Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.

[2] Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.

[3] Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.

[4] Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.

[5] Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.

7.8 External links Measurable function at Encyclopedia of Mathematics

Borel function at Encyclopedia of Mathematics

https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Lp_spacehttps://en.wikipedia.org/wiki/Measure-preserving_dynamical_systemhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-7637-1497-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-49756-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-31716-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-02-404151-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-00754-2http://www.encyclopediaofmath.org/index.php/Measurable_functionhttp://www.encyclopediaofmath.org/http://www.encyclopediaofmath.org/index.php/Borel_functionhttp://www.encyclopediaofmath.org/

Chapter 8

Measure (mathematics)

For the coalgebra concept, see measuring coalgebra.In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of

that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length,area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assignsthe conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclideanspace Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everydaysense of the word, specifically, 1.Technically, a measure is a function that assigns a non-negative real number or + to (certain) subsets of a set X (seeDefinition below). It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset thatcan be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of thesmaller subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfyingthe other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolvedby defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are requiredto form a -algebra. This means that countable unions, countable intersections and complements of measurablesubsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be definedconsistently, are necessarily complicated in the sense of being badly mixed up with their complement.[1] Indeed, theirexistence is a non-trivial consequence of the axiom of choice.Measure theory was developed in successive stages during the late 19th and early 20th centuries by mile Borel,Henri Lebesgue, Johann Radon and Maurice Frchet, among others. The main applications of measures are in thefoundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodictheory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsetsof Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more generaland has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assignto the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

8.1 Definition

Let X be a set and a -algebra over X. A function from to the extended real number line is called a measureif it satisfies the following properties:

Non-negativity: For all E in : (E) 0.

Null empty set: () = 0.

Countable additivity (or -additivity): For all countable collections {Ei}iN of pairwise disjoint sets in :

(k=1

Ek) =k=1

(Ek)

28

https://en.wikipedia.org/wiki/Measuring_coalgebrahttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Lengthhttps://en.wikipedia.org/wiki/Areahttps://en.wikipedia.org/wiki/Volumehttps://en.wikipedia.org/wiki/Euclidean_geometryhttps://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)https://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Real_linehttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Extended_real_number_linehttps://en.wikipedia.org/wiki/Measure_(mathematics)#Definitionhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Countablyhttps://en.wikipedia.org/wiki/Counting_measurehttps://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/%C3%89mile_Borelhttps://en.wikipedia.org/wiki/Henri_Lebesguehttps://en.wikipedia.org/wiki/Johann_Radonhttps://en.wikipedia.org/wiki/Maurice_Fr%C3%A9chethttps://en.wikipedia.org/wiki/Lebesgue_integralhttps://en.wikipedia.org/wiki/Andrey_Kolmogorovhttps://en.wikipedia.org/wiki/Axiomatisationhttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Riemann_integralhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Dynamical_systemhttps://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Extended_real_number_linehttps://en.wikipedia.org/wiki/Sigma_additivityhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Disjoint_sets

8.2. EXAMPLES 29

One may require that at least one set E has finite measure. Then the empty set automatically has measure zero becauseof countable additivity, because (E) = (E ) = (E) + () , so () = (E) (E) = 0 .If only the second and third conditions of the definition of measure above are met, and takes on at most one of theva