sets, numbers, and proofs - division of social sciencesdss.ucsd.edu/~ssaiegh/slides1.pdf · brief...

Brief Refresher on LogicSets

The Set of Real NumbersProofs

Sets, Numbers, and ProofsPOLI 270 - Mathematical and Statistical Foundations

Sebastian M. Saiegh

Department of Political ScienceUniversity California, San Diego

September 23 2010

Sebastian M. Saiegh Sets, Numbers, and Proofs



Sets, Numbers, and Proofs

1 Brief Refresher on LogicDeductive Reasoning and Logical Connectives

2 SetsElements and Subsets

3 The Set of Real NumbersPropertiesThe Continuum Property

4 ProofsWhat is a Proof?Proof Strategies




Overview of Today’s Class

One of the main characteristics of mathematics is its use ofdeductive reasoning to find the answers to questions.

For example, when we solve an equation for x we are usingthe information given by the equation to deduce what thevalue of x must be.

In turn, mathematical analysis starts with the study of the set ofreal numbers and the notion of limit.

These fundamental concepts cannot be introduced withoutmaking reference to concepts such as sets, quantifiers, etc.

Today: A brief exposition of these basic themes and a discussion ofproofs.




Deductive Reasoning and Logical Connectives










Propositional Calculus

Mathematics requires the use of clear and precise language.

These requirements are satisfied by symbolic or mathematicallogic, which gives to every expression an unequivocal meaningand to each symbol an unambiguous interpretation.

A simple way to introduce logical language is given bypropositional calculus.

A Proposition is any expression that is either true or false. In this,propositions differ from questions, commands, and exclamations.





Propositions

Here are two propositions:

“3 is a whole number”

“Charlie is immortal”

The first one is true, and the second one is false.

The expression “x is an even number”, on the other hand, isnot a proposition, because we cannot tell with certaintywhether it is true or false.





Propositions (cont.)

Given the two propositions above, we can obtain otherpropositions. For example:

“3 is a whole number and Charlie is immortal”

“3 is a whole number or Charlie is immortal”

“If 3 is a whole number, then Charlie is immortal”

These new propositions are obtained by compounding the initialones using the words “and”, “or”, “if-then”, etc.

In logic, this connection is done by defining operationsbetween propositions.





Propositions (cont.)

When we construct a new proposition, we may want to establishwether it is true or false for every truth value of the composingones.

The truth value of a true proposition is true and the truthvalue of a false proposition is false.

We usually represent propositions using letters: p, q, r , s, ... and weconstruct tables that give us the necessary information.

These are called truth tables and indicate whether theproposition resulting from an operation is either true or false.





Operations of Propositional Calculus

1 Negation. It is customary to use the symbol “∼” (called atilde) to express negation. So, ∼ p (not p) is a falseproposition if p is true, and true if p is false.

p ∼ p

T FF T

Example: p=“Charlie is immortal” is a false proposition.Thus, its negation ∼ p=“Charlie is not immortal”, is true.





Operations of Propositional Calculus (cont.)

2 Conjunction. It is customary to use the symbol “∧” to expressconjunction (instead of the word “and”). So, the propositionp ∧ q is true only if p and q are both true.

p q p ∧ q

T T TT F FF T FF F F

Example: p=“3 is a whole number” is a true proposition, andq=“Charlie is immortal”, is false. Thus, its conjunction, “3 isa whole number and Charlie is immortal” is a false proposition.






3 Weak Disjunction. It is customary to use the symbol “∨” toexpress disjunction (instead of the word “or”). So, theproposition p ∨ q is false only if p and q are both false.

p q p ∨ q

T T TT F TF T TF F F

Example: p=“3 is a whole number” is a true proposition, andq=“Charlie is immortal”, is false. Thus, its disjunction, “3 isa whole number or Charlie is immortal” is a true proposition.






4 Exclusive Disjunction. In this case, we use the symbol “∨”.So, the proposition p∨q is true only if one of its componentsis true and the other one is false.

p q p∨q

T T FT F TF T TF F F

Example: p=“3 is a whole number” is a true proposition, andq=“Charlie is not immortal”, is true. Thus, its exclusivedisjunction, “3 is a whole number or Charlie is immortal” is afalse proposition.






5 Implication. If p, then q, or p ⇒ q. A proposition of this kindis false only when p is true and q is false.

p q p ⇒ q

T T TT F FF T TF F T

p is called the antecedent, and q is the consequent.Example: p ⇒ q=“If 3 is an even number, then Charlie isimmortal”, is a true proposition.






6 Double Implication. It is customary to use the symbol “⇔”for double implication: p ⇔ q. The proposition p if and onlyif q is true only if p and q have the same truth value.

p q p ⇔ q

T T TT F FF T FF F T

Example: p ⇔ q=“3 is an even number, if and only if Charlieis immortal”, is a true proposition.





Quantifiers

As stated before, the expression “x is an even number” is not aproposition.

However, it becomes a proposition when we replace x with anumber.

We denote such expression as a propositional function of avariable.

A propositional function can be a true proposition for somevalues of the variable, for all of them or for none of its values.

It is convenient to introduce some symbols to account for thesepossibilities.





Quantifiers (cont.)

If the propositional function is a true proposition for all the valuesof the variable, it is customary to indicate this with the “universalquantifier,” symbolized by ∀ (which reads “for all”).

∀x : x is mortal, means that for any meaningful value of x ,the proposition that we will obtain will be true.

If the propositional function is a true proposition for at least someof the values of the variable, it is customary to use the “existentialquantifier,” symbolized by ∃ (which reads “there exists at least onex such that”).

∃x/x is a rectangle, reads as “there exists x such that x is arectangle” and means that there is at least one substitutionfor x which transforms the propositional function into a trueproposition.





Quantifiers (cont.)

All the propositional functions mentioned thus far have had onlyaffirmative singular propositions as substitution instances. But notall propositions are affirmative.

To deny a universal proposition we just have to find a case forwhich it becomes false, that is, we have to find acounterexample.

For example, the universal proposition ∀x : x is mortal isdenied by the existential proposition ∃x/x is not mortal.∼ [∀x : p(x)] ⇔ ∃x/ ∼ p(x)

Similarly, if we want to deny that a proposition is valid for somevalue of x , we need to show that it is false for all values of x :∼ [∃x/p(x)] ⇔ ∀x :∼ p(x).




Elements and Subsets










Sets and Elements

A set is a collection of objects which are called its elements.

The collection of objects must be well-defined.The objects must be distinct (no object in the set is countedtwice).

We usually use upper-case letters such as A,B,C ... to denotesets, and lower-case letters, such as x , y , z ... to designate itselements.

If x is an element of the set S , we say that x belongs to Sand writex ∈ S .

To denote that x is not an element of the set S , we use thenotationx 6∈ S .





Subsets

Suppose that every element of the set S also belongs to theset T , that is, x ∈ S implies that x ∈ T . Then we say that Sis a subset of T and writeS ⊆ T , or T ⊇ S .

Two sets S and T are said to be equal and we write S = T ifand only if they have exactly the same elements. If one of thesets has an element not in the other, they are unequal and wewrite S 6= T .

Equivalently, two sets are said to be equal if each one of themis contained in the other,S = T iff S ⊆ T , or T ⊆ S .





Subsets (cont.)

The statement S ⊆ T does not exclude the possibility thatS = T . In fact, the definition of subset implies that a set is asubset of itself, so S ⊆ S is always true.

However, if S ⊆ T and S 6= T , then we refer to S as aproper subset of T (which is sometimes written as S ⊂ T ).

Suppose that every element of the set S belongs to the set T ,and every element in the set T belongs to the set W . Then,clearly, every element in S belongs to W . In other words, ifS ⊆ T and T ⊆W , then S ⊆W .





Specification of Sets

The simplest way of specifying a set is by listing its elementsand enclosing them in braces. We use the notationA = {1

2 , 1,√

2, π}to denote the set whose elements are the real numbers 1

2 , 1,√2, and π.

Similarly,B = {Romeo, Juliet}denotes the set whose elements are Romeo and Juliet.

For example, given the sets P = {1, 2, 3, 4} and Q = {2, 4}.Then Q ⊆ P. Note that this is not the same thing as writingQ ∈ P, which means that Q is an element of P. The elementsof P are simply 1,2,3, and 4. But Q is not one of these.





Specification of Sets (cont.)

Some sets are not finite, though. Hence, this notation is of nouse when we want to specify a set which has an infinitenumber of elements.

Such sets may be specified by naming the property whichdistinguishes elements of the set from objects which are not inthe set. For example, the notationB = {x : x is an even integer number, x > 0}(which reads as “the set of all x such that x is an even integernumber and x > 0”), denotes the set of all positive eveninteger numbers.

Similarly, D = {y |y loves Romeo} denotes the set of allpeople who love Romeo.





Notation

Braces are always used when specifying a set

We use the colon : and a vertical bar | interchangeably tomean “such that” or “for which”.

The symbol x is just a place-holder; any other symbol will dojust as well.





Universal Set

Unless otherwise stated, all sets under investigation areassumed to be subsets of some fixed set called the universalset and denoted by U.

Once having decided on the universal set for a particulardiscussion, all other sets in that same discussion must besubsets of U. But, different universal sets can be used fordifferent discussions.

For example, in human population studies, the universal setconsists of all the people in the world. But, if we are studyingvoting behavior in the United States, our universal set wouldconsist of all eligible voters in this particular country.





Empty Set

Given a universal set U and a defining property, it may be thecase that there are no elements in U which conform to suchproperty.

For example, if x denotes a variable which ranges over the setof all real numbers, then{x : x2 + 1 = 0}has no elements, because no positive integer can satisfy therequired property (i.e. there are no real numbers x such thatx2 = −1).





Empty Set (cont.)

It is convenient, thus, to have a notation for the empty set ∅.This is the set which has no elements.

Notice that there is only one empty set. If S and T are bothempty, then S = T because both have exactly the sameelements, namely, none.

The empty set is also regarded as a subset of every other set.





Venn Diagrams

A Venn diagram (named after the English logician John Venn,1834-1883) is a graphical representation of sets by simpleplane areas.The universal set U is represented by the area enclosed by arectangle, and the other sets are represented by circles insidethat rectangle.For example, if A ⊆ B, then the circle representing A will becompletely inside the circle representing B.





Venn Diagrams (cont.)

If A and B are disjoint, then the circle that represents A willbe separated from the circle that represents B.





Set Operations: Union and Intersection

Let A and B be arbitrary sets. The union of A and B, denotedby A ∪ B, is the set of elements which belong to A or B:A ∪ B = {x : x ∈ A or x ∈ B}Here “or” is used in the (inclusive) sense of and/or.

The intersection of A and B, denoted by A ∩ B, is the set ofelements which belong to both A and B:A ∩ B = {x : x ∈ A and x ∈ B}If A ∩ B = ∅, that is if A and B do not have any elements incommon, then A and B are said to be disjoint.





Set Operations: Union and Intersection (cont.)

The following properties of union and intersection shall benoted:

(i) Every element x in A ∩ B belongs to A and B. Thus, A ∩ B isa subset of A and of B:A ∩ B ⊆ A and A ∩ B ⊆ B

(ii) An element x belongs to the set A ∪ B if x belongs to A or xbelongs to B; thus, every element in A belongs to A ∪ B, andevery element in B belongs to A ∪ B:A ⊆ A ∪ B and B ⊆ A ∪ B





Set Operations: Complements

Recall that we take all the sets under consideration to besubsets of a particular universal set U.

The absolute complement or, simply, complement of A,denoted by A ′, is the set of elements which do not belong toA:A ′ = {x : x ∈ U, x 6∈ A}Sometimes the complement of A is represented by Ac or A.





Set Operations: Difference and Symmetric Difference

The difference of A and B or the relative complement of Bwith respect to A, denoted by A \ B, is the set of elementswhich belong to A but not to B:A \ B = {x : x ∈ A, x 6∈ B}The set A \ B reads as “A minus B.” Sometimes it isrepresented by A− B or A ∼ B.

The symmetric difference of the sets A and B, denoted byA⊕ B, is the set of elements which belong to A or B, but notto both:A⊕ B = (A ∪ B) \ (A \ B) or A⊕ B = (A \ B) ∪ (B \ A).





Set Operations: Example

Example

Let U = N = {1, 2, 3, ...} be the universal set, andA = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9},E = {2, 4, 6, ...}(E is the set of positive even integers).

Then,

A ′ = {5, 6, 7, ...}, B ′ = {1, 2, 8, 9, 10, ...}, E ′ = {1, 3, 5, ...}(E ′ is the set of positive odd integers).Also, A \ B = {1, 2}, B \ A = {5, 6, 7}, A \ E = {1, 3},E \ A = {6, 8, 10, ...}.And, A⊕ B = (A \ B) ∪ (B \ A) = {1, 2, 5, 6, 7},A⊕ E = {1, 3, 6, 8, 10, ...}.





Classes of Sets

Frequently the members of a set are sets themselves.

For example, each line in a set of lines is a set of points.

To help clarify these situations, we usually use the word class orfamily for such a set.

The words subclass and subfamily have meanings analogousto subset.

Example

The members of the class {{2, 3}, {2}, {5, 6}} are the sets{2, 3}, {2}, and {5, 6}.





Classes of Sets (cont.)

Power Set

Consider any set A. The power set of A, denoted by P(A), isthe class of all subsets of A.

Example

Consider A = {a, b, c}, thenP(A) = {A, {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, ∅}






Partitions

A partition of a set X is a subdivision of X into nonemptysubsets which are disjoint and whose union is X :

A partition is a class of nonempty subsets of X such that eacha ∈ X belongs to a unique subset. The subsets in a partitionare called cells.






Partitions

Example

Consider the following classes of subsets of X = {1, 2, 3, ..., 8, 9}:

(i) [{1, 3, 5}, {2, 6}, {4, 8, 9}](ii) [{1, 3, 5}, {2, 4, 6, 8}, {5, 7, 9}](iii) [{1, 3, 5}, {2, 4, 6, 8}, {7, 9}]

Then (i) is not a partition of X since 7 ∈ X but 7 does not belongto any of the cells. Furthermore, (ii) is not a partition of X since5 ∈ X and 5 belongs to both {1, 3, 5} and {5, 7, 9}. On the otherhand, (iii) is a partition of X since each element of X belongs toexactly one cell.





Special Symbols

Some sets are used so frequently in mathematics, that we havesome special symbols to refer to them. Some of these symbols are:

Therefore, we have N ⊆ Z ⊆ Q ⊆ R.





The Real Line

The set of all real numbers R plays an important role in probabilitytheory, as they are used to convey numerical information. It will beadequate for this class to think of the real numbers as being pointsalong a straight line which extends indefinitely in both directions.The line may be then regarded as an ideal ruler with which we maymeasure the lengths of line segments in Euclidean geometry.




PropertiesThe Continuum Property










Properties: Assumptions

The following assumptions (and their consequences) are concernedwith the properties of the real number system:

Arithmetic. The first assumption is that the real numberssatisfy all the usual laws of addition, substraction,multiplication and division.

Inequalities. The next assumptions concern inequalitiesbetween real numbers and their manipulation.





Inequalities

We assume that, given any two real numbers a and b, there arethree mutually exclusive possibilities:

(i) a > b (a is greater than b)

(ii) a = b (a equals b)

(iii) a < b (a is less than b).

Observe that a < b means the same thing as b > a. We have, forexample, the following inequalities:1 > 0; 3 > 2; 2 < 3; −1 < 0.





Inequalities (cont.)

There is often some confusion about the statements

(iv) a ≥ b (a is greater than or equal to b)

(v) a ≤ b (a is less than or equal to b).

To clear up this confusion, we note that the following are all truestatements.1 ≥ 0; 3 ≥ 2; 1 ≥ 1; 2 ≤ 3; −1 ≤ 0; −3 ≤ −3.





Inequalities (cont.)

We assume four basic rules for the manipulation of inequalities:

(I) If a > b and b > c , then a > c .

(II) If a > b and c is any real number, thena + c > b + c .

(III) If a > b and c > 0, then ac > bc (i.e. inequalities can bemultiplied through by a positive factor).

(IV) If a > b and c < 0, then ac < bc (i.e. multiplication by anegative factor reverses the inequality).





Roots

Let n be a natural number. You guys should be familiar with thenotation y = xn.

For example, x2 = x � x and x3 = x � x � x .

Our next assumption about the real number system is thefollowing. Given any y ≥ 0 there is exactly one value of x ≥ 0 suchthaty = xn.

If y ≥ 0, the value of x ≥ 0 which satisfies the equationy = xn is called the nth root of y and is denoted by

x = y1n .





Roots (cont.)

When n = 2, we also use the notation√

y = y12 .

Note that with this convention, it is always true that√

y ≥ 0.

If y > 0, there are, of course, two numbers whose square is y .

The positive one is√

y and the negative one is −√y .

The notation ±√y means “√

y or −√y”.





Roots (cont.)

If r = mn is a positive rational number and y ≥ 0, we define

y r = (ym)1n .

If r is a negative rational, then −r is a positive rational andhence y−r is defined. If y > 0 we can therefore define y r byy r = 1

y−r .

We also write y0 = 1. With these conventions it follows that, ify > 0, then y r is defined for all rational numbers r .





Quadratic equations

If y > 0, the equation x2 = y has two solutions.

We denote the positive solution by√

y .

The negative solution is therefore −√y .

The general quadratic equation has the formax2 + bx + c = 0where a 6= 0.





Quadratic equations (cont.)

Multiply through by 4a. We obtain

4a2x2 + 4abx + 4ac = 0

(2ax + b)2 − b2 + 4ac = 0

(2ax + b)2 = b2 − 4ac

It follows that the quadratic equation has no real solutions ifb2 − 4ac < 0, one real solution if b2 − 4ac = 0 and two realsolutions if b2 − 4ac > 0.





Quadratic equations (cont.)

If b2 − 4ac ≥ 0,

2ax + b = ±√

(b2 − 4ac)

x =−b ±

√(b2 − 4ac)

2a.

The roots of the equation ax2 + bx + c = 0 are therefore

α =−b −

√(b2 − 4ac)

2a,

and

β =−b +

√(b2 − 4ac)

2a.





Modulus

Suppose that x is a real number.

Its modulus (or absolute value) |x | is defined by

|x | =

{x if x ≥ 0−x if x < 0.

Therefore |3| = 3, | − 6| = 6 and |0| = 0.

Obviously, |x | ≥ 0 for all values of x .

Note also that |x | =√

x2.





Brad Pitt (a.k.a Achilles) and the Tortoise

The fifth century B.C. Greek philosopher Zeno of Elia inventedseveral famous paradoxes.

The following is one of the most famous: Achilles is to race atortoise.

Since Achilles runs faster than the tortoise, the tortoise isgiven a start of x0 feet.

When Achilles reaches the point where the tortoise started,the tortoise will have advanced a bit, say x1 feet.





Brad Pitt (a.k.a Achilles) and the Tortoise (cont.)

Achilles soon reaches the tortoise’s new position, but, by then, thetortoise will have advanced a little bit more, say x2 feet.

This argument may be continued indefinitely and so Achillescan never catch the tortoise.






The simplest way to resolve this paradox is to say that Achillescatches the tortoise after he has run a distance of x feet, where xis ‘the smallest real number larger than all of the numbersx0, x0 + x1, x0 + x1 + x2,...’

Zeno’s argument then simply reduces to subdividing a linesegment of length x into an infinite number of smaller linesegments of respective lengths x0, x1, x2...






Formulated in this way, the paradox losses its sting.

This solution, though, depends very strongly on the existenceof the real number x (i.e. the smallest real number largestthan all the numbers x0, x0 + x1, x0 + x1 + x2,...)





Bounds

A set S of real numbers is bounded above if there exists a realnumber k which is greater than or equal to every element of theset.

For some k ,x ≤ k

for any x ∈ S .

The number k (if such number exists) is called an upper bound ofthe set S .





Bounds (cont.)

A set S of real numbers is bounded below if there exists a realnumber h which is less than or equal to every element of the set.

For some h,x ≥ h

for any x ∈ S .

The number h (if such number exists) is called an lower bound ofthe set S .





Bounds (cont.)

A set which is both bounded above and bounded below is just saidto be bounded.

Definition

A set S of real numbers is bounded if and only if there exists a realnumber k such that|x | ≤ kfor any x ∈ S .





Bounds (cont.)

Example

The set {x : 1 ≤ x < 2} is bounded above. Some upper boundsare 100,10,4, and 2. The set is also bounded below. Some lowerbounds are -27,0, and 1.

Example

The set {x : x > 0} is unbounded above. If h > 0 is proposed as anupper bound, one has only to point to h + 1 to obtain an elementof the set larger than the supposed upper bound. However, the set{x : x > 0} is bounded below. Some lower bounds are -27, and 0.





Continuum Property

Definition

Continuum Property. Every non-empty set of real numbers whichis bounded above has a smallest upper bound. Every non-empty setof real numbers which is bounded below has a largest lower bound.





Continuum Property (cont.)

Therefore, if S is a non-empty set which is bounded above, then Shas an upper bound b such that, given any other upper bound k ofS ,b ≤ k.

Similarly, if S is a non-empty set which is bounded below,then S has an lower bound c such that, given any other lowerbound h,c ≥ h.





Continuum Property (cont.)

Example

The smallest upper bound of the set {x : 1 ≤ x < 2} is 2. Thelargest lower bound is 1.

Example

The set {x : x > 0} has no upper bounds at all. The largest lowerbound of this set is 0.





Supremum and Infimum

If a non-empty set S is bounded above, then, by the ContinuumProperty, it has a smallest upper bound k .

This smallest upper bound k is sometimes called thesupremum of the set S . We write k = sup S or

k = supx∈S

x .

Similarly, a set which is bounded below has a largest lowerbound b. We call b the infimum of the set S and writeb = inf S or

b = infx∈S

x .





Supremum and Infimum (cont.)

Sometimes you guys may see the notation sup S = +∞.

This simply means that S is unbounded above.

Similarly, if inf S = −∞ means that S is unbounded below.





Maximum and Minimum

If a set S has a largest element m, we call m the maximum of theset S and write m = max S .

If S has a smallest element n, we call n the minimum of Sand write n = min S .

It is fairly obvious that, if a set S has a maximum m, then itis bounded above and its smallest upper bound is m. Thus, inthis case, sup S = max S .





Maximum and Minimum (cont.)

Example

The set {1, 2, 3} has a maximum 3 and this is equal to its smallestupper bound. The set {1, 2, 3} has a minimum 1 and this is equalto its largest lower bound.

A common error, though, is to suppose that the smallest upperbound of a set S is always the maximum of the set. Some setswhich are bounded above (and hence have a smallest upperbound) do not have a maximum.





Maximum and Minimum (cont.)

Example

The set {x : 1 ≤ x < 2} has no maximum. The number 2 cannotbe the largest element of the set because it does not belong to it.On the other hand, any x in the set satisfies 1 ≤ x < 2. But theny = x+2

2 is an element of the set which is larger than x . Hence, xcannot be the largest element of the set. The set has a smallestupper bound 2.

Example

The set {x : x > 0} has no maximum, nor does it have any upperbounds. The set {x : x > 0} has no minimum. Its largest lowerbound is 0.





Intervals

An interval I is a set of real numbers with the property that, ifx ∈ I and y ∈ I and x ≤ z ≤ y , then z ∈ I .

In words, if two numbers belong to I , then so does everynumber between them.





Intervals (cont.)

In describing intervals we use the following notation:

(a, b) = {x : a < x < b}[a, b] = {x : a ≤ x ≤ b}[a, b) = {x : a ≤ x < b}(a, b] = {x : a < x ≤ b}.

These are the bounded intervals (classified by wether or not theyhave a maximum and whether or not they have a minimum).





Intervals (cont.)

We also need to consider the unbounded intervals. For these weuse the following notation:

(a,∞) = {x : x > a}[a,∞] = {x : x ≥ a}(−∞, b) = {x : x < b}(−∞, b] = {x : x ≤ b}.

Please pay special attention to the symbols ∞ and −∞, andremember that we use them for notational convenience and not asreal numbers.





Intervals (cont.)

All intervals (with the exception of the empty set and the set of allreal numbers) fall into one of the categories described above.

We call the intervals (a, b), (a,∞) and (−∞, b) openintervals.

We call the intervals [a, b], [a,∞) and (−∞, b] closedintervals.

We call the closed, bounded intervals [a, b], compact intervals.




What is a Proof?Proof Strategies

Deductive Reasoning

Now that we have covered the basics on deductive reasoning andthe language of mathematics, it is the turn to put this knowledgeinto practice.

One of the important roles of mathematics in sciences is todeduce complex scientific principles from a collection ofgenerally agreed assumptions.

The goal is to arrive at a conclusion from the assumption thatsome other statements, called premises, are true.

Deductive reasoning in mathematics is usually presented in theform of a proof.





Statements

In mathematics, a statement is a sentence expressed in words (ormathematical symbols) that is either true or false. Examples:

1. 1+1=2.

2. x2 + 1 = 0 (x is a positive integer).

3. 3x = 5 and y = 1

4. Given two real numbers a and b, if 0 < a < b, then a2 < b2.

Note that statement (1) is always true, (2) is always false, andstatement (3) is either true or false, depending on the value of avariable. For this reason, (3) is called a conditional statement.





Proven Statements

It is perhaps not as obvious that statement (4) is always true.

It therefore becomes necessary to have some method forproving that such statements are true.

A proof is a logical argument that establishes the truth of astatement beyond ay doubt. In other words, a proof is a convincingargument expressed in the language of mathematics.





Proving Statements

Going back to (4), the task is the following: given two statementsA and B, each of which may be either true or false, show that thefollowing statement – called an implication – is true:

If A is true, then B is true.

How do we prove it? First, we start with statement A(0 < a < b) – called the hypothesis (a statement that it isassumed to be true, and from which some consequencefollows)

The consequence, in this case is B (a2 < b2). This statementis called the conclusion: (it follows from previously assumedconditions – hypotheses).





Proving Statements (cont.)

Recall from the beginning of today’s class, that implication is oneof the operations of propositional calculus.

You may also remember that a general feature of statementsof the form A implies B is that there is only one case in whichthe statement is false: when A is true and B is false.

Therefore, we can prove that a statement of the form “If A, thenB” is true if it is impossible for A to be true and B to be false atthe same time; that is, whenever A is true, B must be true as well.





Proof Techniques

Just as there are many ways to express the same idea in language,so there are different ways of proving the same mathematical fact.

A proof should contain enough mathematical details to beconvincing to the person(s) to whom it is addressed.

We will now categorize and explain the various proof techniquesthat are used in all proofs, regardless of the subject matter.





Direct Proof

A direct proof is based on the assumption that the hypothesiscontains enough information to allow the construction of a seriesof logically connected steps leading to the conclusion.

Note, that from this point of view, a proof of the statement Aimplies B is not an attempt to verify whether A and Bthemselves are true but rather to show that B is a logicalresult of having assumed that A is true.

We can now go back and prove statement (4).





Direct Proof (cont.)

Example

Suppose a and b are real numbers. Prove that if 0 < a < b thena2 < b2.

We are given as an ancillary hypothesis the statement that a and bare real numbers. Our problem has the from A implies B where Ais the statement 0 < a < b and B is the statement a2 < b2.

According to this proof technique, we should assume that0 < a < b and try to use this assumption to prove thata2 < b2.






Comparing the inequalities a < b and a2 < b2 suggests thatmultiplying both sides of the given inequality a < b by either a orb might get us closer to proving the statement.

Because we are given that a and b are positive, we won’t needto reverse the direction of the inequality if we do this.

Multiplying a < b by a gives us a2 < ab, and multiplying it byb gives us ab < b2.

Therefore a2 < ab < b2, so a2 < b2, as required.

Thus, if 0 < a < b then a2 < b2.






Notice that in the course of proving statement (4) we implicitlyused a lot of mathematical knowledge such as for example, theproperties of multiplication.

These properties, in turn, depend on statements that havebeen proved using other statements and so on and so forth.

More generally, then, we can characterize the direct way of provingthat A⇒ B as the process of finding a sequence of acceptedaxioms and theorems of the form Ai ⇒ Ai+1 for i = 1, ..., n sothat A0 = A and An+1 = B.





Theorems

Definition

A theorem is a mathematical statement for which the truth canbe established using logical reasoning on the basis of certainassumptions that are explicitly given or implied in the statement(i.e. by constructing a proof).





Theorems (cont.)

Theorem

For any x , y , z ∈ R, if x + z = y + z then x = y.

Proof.

1. x + z = y + z is our hypothesis

2. Using the additive inverse property, we can establish thatthere exists (−z) such that z + (−z) = 0.

3. Using again the properties of addition and multiplication, weget (x + z) + (−z) = (y + z) + (−z).

4. Finally, using the additive associative property, it must be truethat x + (z + (−z)) = y + (z + (−z)).

5. Given step (2), then x + 0 = y + 0.

6 By the additive identity property, x = y .





Theorems (cont.)

As this example show, the theorem says that if certain assumptions(the hypotheses) are true, then some conclusion must also be true.

Yet, often the hypotheses and conclusion contain freevariables.

An assignment of particular values to these variables is called aninstance of the theorem.

If there is even one instance in which the hypotheses are truebut the conclusion is false, then the theorem is incorrect.Such an instance is called a counterexample to the theorem.





Indirect Proof

There is a second method that is sometimes used for provingstatements of the form A⇒ B.

Because any conditional statement A⇒ B is equivalent to itscontrapositive ∼ B ⇒∼ A, this strategy consists in provingA⇒ B by proving ∼ B ⇒∼ A.

This gives us a different starting point because we will start byassuming that B is false, and we will prove that this impliesthat A is false, as the contrapositive of the original statementis “If ‘not B,’ then ‘not A.’ ”





Indirect Proof (cont.)

Example

Given the inequality a > 0, we are going to prove that a−1 > 0.

Proof. We are going to prove this by contradiction. Suppose thata > 0, but that a−1 ≤ 0. It cannot be true that a−1 = 0 (sincethen 0 = 0 � a = 1). Hence, a−1 < 0.

Recall that inequalities can be multiplied through by a positivefactor. In this case, we can multiply the previous inequalitythrough by a (since a > 0).

Hence, 1 = a−1 � a < 0 � a = 0. But 1 < 0 is a contradiction.Therefore the assumption a−1 ≤ 0 was false.

Hence a−1 > 0.





Indirect Proof (cont.)

Example

Suppose that, for any ε > 0, a < b + ε. Then, a ≤ b.

Proof. Assume that a > b. Then a− b > 0. But, for any ε > 0,a < b + ε.

Hence, a < b + ε in the particular case when ε = a− b. Thus,a < b + (a− b) and so a < a.

Contradiction! Hence our assumption a > b must be false.Therefore, a ≤ b.


sets, numbers, and proofs - division of social sciencesdss.ucsd.edu/~ssaiegh/slides1.pdf · brief...

Documents