discrete computational structures cse 2353 fall 2011 most slides modified from discrete mathematical...
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DISCRETE COMPUTATIONAL STRUCTURES
CSE 2353
Fall 2011Most slides modified from
Discrete Mathematical Structures: Theory and Applications
CSE 2353 OUTLINE
1. Sets 2. Logic3. Proof Techniques4. Integers and Induction5. Relations and Posets6. Functions7. Counting Principles8. Boolean Algebra
CSE 2353 OUTLINE
1.Sets 2. Logic3. Proof Techniques4. Integers and Induction5. Relations and Posets6. Functions7. Counting Principles8. Boolean Algebra
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Sets: Learning Objectives
Learn about sets
Explore various operations on sets
Become familiar with Venn diagrams
CS:
Learn how to represent sets in computer memory
Learn how to implement set operations in programs
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Sets
Definition: Well-defined collection of distinct objectsMembers or Elements: part of the collectionRoster Method: Description of a set by listing the
elements, enclosed with bracesExamples:
Vowels = {a,e,i,o,u}Primary colors = {red, blue, yellow}
Membership examples “a belongs to the set of Vowels” is written as: a Vowels “j does not belong to the set of Vowels: j Vowels
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Sets
Set-builder method
A = { x | x S, P(x) } or A = { x S | P(x) }
A is the set of all elements x of S, such that x satisfies the property P
Example:
If X = {2,4,6,8,10}, then in set-builder notation, X can be described as
X = {n Z | n is even and 2 n 10}
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Sets Standard Symbols which denote sets of numbers
N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z+ : The set of all positive integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* : The set of all nonzero rational numbers Q+ : The set of all positive rational numbers R : The set of all real numbers R* : The set of all nonzero real numbers R+ : The set of all positive real numbers C : The set of all complex numbers C* : The set of all nonzero complex numbers
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Sets
Subsets
“X is a subset of Y” is written as X Y
“X is not a subset of Y” is written as X Y
Example:
X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g}
Y X, since every element of Y is an element of X
Y Z, since a Y, but a Z
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Sets
SupersetX and Y are sets. If X Y, then “X is contained in
Y” or “Y contains X” or Y is a superset of X, written Y X
Proper SubsetX and Y are sets. X is a proper subset of Y if X
Y and there exists at least one element in Y that is not in X. This is written X Y.
Example: X = {a,e,i,o,u}, Y = {a,e,i,o,u,y}
X Y , since y Y, but y X
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Sets Set Equality
X and Y are sets. They are said to be equal if every element of X is an element of Y and every element of Y is an element of X, i.e. X Y and Y X
Examples:{1,2,3} = {2,3,1}X = {red, blue, yellow} and Y = {c | c is a primary
color} Therefore, X=Y
Empty (Null) SetA Set is Empty (Null) if it contains no elements.The Empty Set is written as The Empty Set is a subset of every set
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Sets
Finite and Infinite SetsX is a set. If there exists a nonnegative integer n
such that X has n elements, then X is called a finite set with n elements.
If a set is not finite, then it is an infinite set.
Examples: Y = {1,2,3} is a finite set
P = {red, blue, yellow} is a finite set
E , the set of all even integers, is an infinite set
, the Empty Set, is a finite set with 0 elements
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Sets
Cardinality of Sets
Let S be a finite set with n distinct elements, where n ≥ 0. Then |S| = n , where the cardinality (number of elements) of S is n
Example:If P = {red, blue, yellow}, then |P| = 3
Singleton A set with only one element is a singleton
Example:H = { 4 }, |H| = 1, H is a singleton
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Sets
Power Set
For any set X ,the power set of X ,written P(X),is the set of all subsets of X
Example:
If X = {red, blue, yellow}, then P(X) = { , {red}, {blue}, {yellow}, {red,blue}, {red, yellow}, {blue, yellow}, {red, blue, yellow} }
Universal Set
An arbitrarily chosen, but fixed set
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Sets
Venn DiagramsAbstract visualization of
a Universal set, U as a rectangle, with all subsets of U shown as circles.
Shaded portion represents the corresponding set
Example: In Figure 1, Set X,
shaded, is a subset of the Universal set, U
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Set Operations and Venn Diagrams
Union of Sets
Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then XUY = {1,2,3,4,5,6,7,8,9}
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Sets
Intersection of Sets
Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5}
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Sets
Disjoint Sets
Example: If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y =
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Sets
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Sets
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Sets
Difference
• Example: If X = {a,b,c,d} and Y = {c,d,e,f}, then X – Y = {a,b} and Y – X = {e,f}
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Sets
Complement
Example: If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b}
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Sets
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Sets
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Sets
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SetsOrdered Pair
X and Y are sets. If x X and y Y, then an ordered pair is written (x,y)
Order of elements is important. (x,y) is not necessarily equal to (y,x)
Cartesian ProductThe Cartesian product of two sets X and Y ,written
X × Y ,is the setX × Y ={(x,y)|x ∈ X , y ∈ Y}
For any set X, X × = = × XExample:
X = {a,b}, Y = {c,d} X × Y = {(a,c), (a,d), (b,c), (b,d)}Y × X = {(c,a), (d,a), (c,b), (d,b)}
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Computer Representation of Sets
A Set may be stored in a computer in an array as an unordered listProblem: Difficult to perform operations on the set.
Linked ListSolution: use Bit Strings (Bit Map)
A Bit String is a sequence of 0s and 1sLength of a Bit String is the number of digits in the
stringElements appear in order in the bit string
A 0 indicates an element is absent, a 1 indicates that the element is present
A set may be implemented as a file
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Computer Implementation of Set Operations
Bit Map
File
OperationsIntersection
Union
Element of
Difference
Complement
Power Set
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Special “Sets” in CS
Multiset
Ordered Set
CSE 2353 OUTLINE
1. Sets 2.Logic
3. Proof Techniques4. Relations and Posets
5. Functions6. Counting Principles
7. Boolean Algebra
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Logic: Learning Objectives
Learn about statements (propositions)
Learn how to use logical connectives to combine statements
Explore how to draw conclusions using various argument forms
Become familiar with quantifiers and predicates
CS
Boolean data type
If statement
Impact of negations
Implementation of quantifiers
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Mathematical Logic
Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid
Theorem: a statement that can be shown to be true (under certain conditions)
Example: If x is an even integer, then x + 1 is an odd integer
This statement is true under the condition that x is an integer is true
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Mathematical Logic
A statement, or a proposition, is a declarative sentence that is either true or false, but not both
Lowercase letters denote propositionsExamples:
p: 2 is an even number (true)
q: 3 is an odd number (true)
r: A is a consonant (false)
The following are not propositions:p: My cat is beautiful
q: Are you in charge?
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Mathematical Logic Truth value
One of the values “truth” (T) or “falsity” (F) assigned to a statement
NegationThe negation of p, written ~p, is the statement obtained by
negating statement p Example:
p: A is a consonant~p: it is the case that A is not a consonant
Truth Table
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Mathematical Logic
ConjunctionLet p and q be statements.The
conjunction of p and q, written p ^ q , is the statement formed by joining statements p and q using the word “and”
The statement p ^ q is true if both p and q are true; otherwise p ^ q is false
Truth Table for Conjunction:
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Mathematical Logic
DisjunctionLet p and q be statements. The disjunction of p
and q, written p v q , is the statement formed by joining statements p and q using the word “or”
The statement p v q is true if at least one of the statements p and q is true; otherwise p v q is false
The symbol v is read “or”
Truth Table for Disjunction:
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Mathematical Logic Implication
Let p and q be statements.The statement “if p then q” is called an implication or condition.
The implication “if p then q” is written p q
“If p, then q””p is called the hypothesis, q is called the
conclusionTruth Table for
Implication:
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Mathematical Logic
ImplicationLet p: Today is Sunday and q: I will wash the car. p q :
If today is Sunday, then I will wash the carThe converse of this implication is written q p
If I wash the car, then today is SundayThe inverse of this implication is ~p ~q
If today is not Sunday, then I will not wash the carThe contrapositive of this implication is ~q ~p
If I do not wash the car, then today is not Sunday
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Mathematical Logic
BiimplicationLet p and q be statements. The statement “p if and
only if q” is called the biimplication or biconditional of p and q
The biconditional “p if and only if q” is written p q“p if and only if q”Truth Table for the
Biconditional:
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Mathematical Logic
Statement Formulas Definitions
Symbols p ,q ,r ,...,called statement variables
Symbols ~, ^, v, →,and ↔ are called logical
connectives1) A statement variable is a statement formula2) If A and B are statement formulas, then the
expressions (~A ), (A ^ B) , (A v B ), (A → B )
and (A ↔ B ) are statement formulas Expressions are statement formulas that are
constructed only by using 1) and 2) above
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Mathematical Logic
Precedence of logical connectives is:
~ highest
^ second highest
v third highest
→ fourth highest
↔ fifth highest
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Mathematical Logic
Tautology
A statement formula A is said to be a tautology if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A
Contradiction
A statement formula A is said to be a contradiction if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A
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Mathematical Logic
Logically ImpliesA statement formula A is said to logically imply a
statement formula B if the statement formula A → B is a tautology. If A logically implies B, then symbolically we write A → B
Logically EquivalentA statement formula A is said to be logically
equivalent to a statement formula B if the statement formula A ↔ B is a tautology. If A is logically equivalent to B , then symbolically we write A ≡ B
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Mathematical Logic
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Validity of Arguments
Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion
Argument: a finite sequence of statements.
The final statement, , is the conclusion, and the statements are the premises of the argument.
An argument is logically valid if the statement formula is a tautology.
AAAAA nn,...,,,,
1321
An
AAAA n 1321...,,,,
AAAAA nn
1321...,,,,
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Validity of Arguments
Valid Argument FormsModus Ponens:
Modus Tollens :
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Validity of Arguments
Valid Argument FormsDisjunctive Syllogisms:
Hypothetical Syllogism:
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Validity of ArgumentsValid Argument Forms
Dilemma:
Conjunctive Simplification:
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Validity of Arguments
Valid Argument FormsDisjunctive Addition:
Conjunctive Addition:
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Quantifiers and First Order Logic
Predicate or Propositional Function
Let x be a variable and D be a set; P(x) is a sentence
Then P(x) is called a predicate or propositional function with respect to the set D if for each value of x in D, P(x) is a statement; i.e., P(x) is true or false
Moreover, D is called the domain of the discourse and x is called the free variable
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Quantifiers and First Order Logic
Universal Quantifier
Let P(x) be a predicate and let D be the domain of the discourse. The universal quantification of P(x) is the statement:
For all x, P(x) or
For every x, P(x)
The symbol is read as “for all and every”
Two-place predicate:
)( xPx),( yxPyx
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Quantifiers and First Order Logic
Existential Quantifier
Let P(x) be a predicate and let D be the domain of the discourse. The existential quantification of P(x) is the statement:
There exists x, P(x)
The symbol is read as “there exists”
Bound VariableThe variable appearing in: or
)( xPx
)( xPx )( xPx
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Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws) Example:
If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at least one runner who has not won a race. Therefore:
and so,
)(~ )( ~ xPxxPx
)( xPx
)(~ xPx
)(~ )( ~ xPxxPx
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Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws)
)(~ )( ~ xPxxPx
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Arguments in Predicate Logic
Universal SpecificationIf is true, then F(a) is true
Universal GeneralizationIf F(a) is true then is true
Existential SpecificationIf is true, then where F(a) is true
Existential GeneralizationIf F(a) is true then is true
)( xFx
Ua )( xFx
)( xFx
Ua
Ua
Ua
)( xFx
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Logic and CS
Logic is basis of ALULogic is crucial to IF statements
ANDORNOT
Implementation of quantifiersLooping
Database Query LanguagesRelational AlgebraRelational CalculusSQL
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques4. Integers and Inductions
5. Relations and Posets6. Functions
7. Counting Principles8. Boolean Algebra
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Proof Technique: Learning Objectives
Learn various proof techniques
Direct
Indirect
Contradiction
Induction
Practice writing proofs
CS: Why study proof techniques?
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Proof Techniques
Theorem
Statement that can be shown to be true (under certain conditions)
Typically Stated in one of three ways
As Facts
As Implications
As Biimplications
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Proof Techniques
Direct Proof or Proof by Direct MethodProof of those theorems that can be
expressed in the form ∀x (P(x) → Q(x)), D is the domain of discourse
Select a particular, but arbitrarily chosen, member a of the domain D
Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true
Show that Q(a) is trueBy the rule of Universal Generalization (UG), ∀x (P(x) → Q(x)) is true
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Proof Techniques
Indirect Proof
The implication p → q is equivalent to the implication (∼q → ∼p)
Therefore, in order to show that p → q is true, one can also show that the implication (∼q → ∼p) is true
To show that (∼q → ∼p) is true, assume that the negation of q is true and prove that the negation of p is true
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Proof Techniques
Proof by Contradiction Assume that the conclusion is not true and then
arrive at a contradictionExample: Prove that there are infinitely many prime
numbersProof:
Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn
Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes
Therefore, q is a prime. However, it was not listed.Contradiction! Therefore, there are infinitely many
primes.
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Proof Techniques
Proof of Biimplications To prove a theorem of the form ∀x (P(x) ↔
Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true
The biimplication p ↔ q is equivalent to (p → q) ∧ (q → p)
Prove that the implications p → q and q → p are trueAssume that p is true and show that q is trueAssume that q is true and show that p is true
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Proof Techniques
Proof of Equivalent Statements
Consider the theorem that says that statements p,q and r are equivalent
Show that p → q, q → r and r → p Assume p and prove q. Then assume q
and prove r Finally, assume r and prove p
What other methods are possible?
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Other Proof Techniques
Vacuous
Trivial
Contrapositive
Counter Example
Divide into Cases
Constructive
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Proof Basics
You can not prove by example
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Proofs in Computer Science
Proof of program correctness
Proofs are used to verify approaches
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques
4. Integers and Induction5. Relations and Posets
6. Functions7. Counting Principles
8. Boolean Algebra
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Learning Objectives
Learn how the principle of mathematical induction is used to solve problems and proofs
Learn about the basic properties of integers
Explore how addition and subtraction operations are performed on binary numbers
CS
Become aware how integers are represented in computer memory
Looping
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Mathematical Deduction
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Mathematical Deduction
Proof of a mathematical statement by the principle of mathematical induction consists of three steps:
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Mathematical DeductionAssume that when a domino is knocked over, the next domino
is knocked over by itShow that if the first domino is knocked over, then all the
dominoes will be knocked over
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Mathematical Deduction
Let P(n) denote the statement that then nth domino is knocked over
Show that P(1) is trueAssume some P(k) is true, i.e. the kth domino is
knocked over for some
Prove that P(k+1) is true, i.e.
1k
)1()( kPkP
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Mathematical Deduction
Assume that when a staircase is climbed, the next staircase is also climbed
Show that if the first staircase is climbed then all staircases can be climbed
Let P(n) denote the statement that then nth staircase is climbed
It is given that the first staircase is climbed, so P(1) is true
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Mathematical Deduction
Suppose some P(k) is true, i.e. the kth staircase is climbed for some
By the assumption, because the kth staircase was climbed, the k+1st staircase was climbed
Therefore, P(k) is true, so
1k
)1()( kPkP
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Mathematical Deduction
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Mathematical Deduction
We can associate a predicate, P(n). The predicate P(n) is such that:
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Integers
Properties of Integers
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Integers
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Integers
The div and mod operatorsdiv
a div b = the quotient of a and b obtained by dividing a on b.
Examples:8 div 5 = 113 div 3 = 4
moda mod b = the remainder of a and b obtained by dividing
a on b8 mod 5 = 313 mod 3 = 1
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Integers
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Integers
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Integers
Relatively Prime Number
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Integers
Least Common Multiples
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Representation of Integers in Computers
Digital Signals0s and 1s – 0s represent low voltage, 1s high
voltageDigital signals are more reliable carriers of
information than analog signalsCan be copied from one device to another
with exact precisionMachine language is a sequence of 0s and 1s
The digit 0 or 1 is called a binary digit , or bit A sequence of 0s and 1s is sometimes referred to
as binary code
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Representation of Integers in Computers
Decimal System or Base-10The digits that are used to represent numbers in base 10
are 0,1,2,3,4,5,6,7,8, and 9 Binary System or Base-2
Computer memory stores numbers in machine language, i.e., as a sequence of 0s and 1s
Octal System or Base-8Digits that are used to represent numbers in base 8 are
0,1,2,3,4,5,6, and 7Hexadecimal System or Base-16
Digits and letters that are used to represent numbers in base 16 are 0,1,2,3,4,5,6,7,8,9,A ,B ,C ,D ,E , and F
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Representation of Integers in Computers
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Representation of Integers in Computers
Two’s Complements and Operations on Binary NumbersIn computer memory, integers are
represented as binary numbers in fixed-length bit strings, such as 8, 16, 32 and 64
Assume that integers are represented as 8-bit fixed-length strings
Sign bit is the MSB (Most Significant Bit) Leftmost bit (MSB) = 0, number is positiveLeftmost bit (MSB) = 1, number is negative
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Representation of Integers in Computers
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Representation of Integers in Computers
One’s Complements and Operations on Binary Numbers
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Representation of Integers in Computers
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Prime Numbers
Example:Consider the integer 131. Observe that 2 does not divide 131. We now find all odd primes p such that p2 131. These primes are 3, 5, 7, and 11. Now none of 3, 5, 7, and 11 divides 131. Hence, 131 is a prime.
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Prime Numbers
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Prime Numbers
Factoring a Positive Integer
The standard factorization of n
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Prime Numbers
Fermat’s Factoring Method
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Prime Numbers Fermat’s Factoring Method
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques4. Integers and Induction
5.Relations and Posets6. Functions
7. Counting Principles8. Boolean Algebra
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Learning Objectives
Learn about relations and their basic properties
Explore equivalence relations
Become aware of closures
Learn about posets
Explore how relations are used in the design of relational databases
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Relations
Relations are a natural way to associate objects of various sets
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Relations
R can be described in
Roster form
Set-builder form
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Relations
Arrow Diagram
Write the elements of A in one column
Write the elements B in another column
Draw an arrow from an element, a, of A to an element, b, of B, if (a ,b) R
Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is defined as follows: For all a A and b B, a R b if and only if a divides b
The symbol → (called an arrow) represents the relation R
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Relations
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Relations
Directed Graph
Let R be a relation on a finite set A
Describe R pictorially as follows:
For each element of A , draw a small or big dot and label the dot by the corresponding element of A
Draw an arrow from a dot labeled a , to another dot labeled, b , if a R b .
Resulting pictorial representation of R is called the directed graph representation of the relation R
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Relations
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RelationsDomain and Range of the Relation
Let R be a relation from a set A into a set B. Then R ⊆ A x B. The elements of the relation R tell which element of A is R-related to which element of B
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Relations
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Relations
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Relations
Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r ), (3, q), (4, p)}. Then R−1 = {(q, 1), (r , 2), (q, 3), (p, 4)}
To find R−1, just reverse the directions of the arrows
D(R) = {1, 2, 3, 4} = Im(R−1), Im(R) = {p, q, r} = D(R−1)
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Relations
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Relations
Constructing New Relations from Existing Relations
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Relations
Example:Consider the relations R and S as given in Figure 3.7.The composition S ◦ R is given by Figure 3.8.
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
Hasse DiagramLet S = {1, 2, 3}. Then
P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}
Now (P(S),≤) is a poset, where ≤ denotes the set inclusion relation. The poset diagram of (P(S),≤) is shown in Figure 3.22
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Partially Ordered Sets
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Partially Ordered Sets
Hasse DiagramLet S = {1, 2, 3}. Then
P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}
(P(S),≤) is a poset, where ≤ denotes the set inclusion relation
Draw the digraph of this inclusion relation (see Figure 3.23). Place the vertex A above vertex B if B ⊂ A. Now follow steps (2), (3), and (4)
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Partially Ordered Sets
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Partially Ordered Sets
Hasse DiagramConsider the poset (S,≤), where
S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation.
2 and 5 are the only minimal elements of this poset.
This poset has no least element.20 and 15 are the only maximal
elements of this poset.This poset has no greatest
element.
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Application: Relational Database
A database is a shared and integrated computer structure that storesEnd-user data; i.e., raw facts that are of interest
to the end user;Metadata, i.e., data about data through which
data are integratedA database can be thought of as a well-organized
electronic file cabinet whose contents are managed by software known as a database management system; that is, a collection of programs to manage the data and control the accessibility of the data
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Application: Relational Database
In a relational database system, tables are considered as relations
A table is an n-ary relation, where n is the number of columns in the tables
The headings of the columns of a table are called attributes, or fields, and each row is called a record
The domain of a field is the set of all (possible) elements in that column
CSE 2353 f11
Application: Relational Database
Each entry in the ID column uniquely identifies the row containing that ID
Such a field is called a primary key
Sometimes, a primary key may consist of more than one field
CSE 2353 f11
Application: Relational Database
Structured Query Language (SQL)Information from a database is retrieved via a
query, which is a request to the database for some information
A relational database management system provides a standard language, called structured query language (SQL)
A relational database management system provides a standard language, called structured query language (SQL)
CSE 2353 f11
Application: Relational Database
Structured Query Language (SQL)An SQL contains commands to create tables,
insert data into tables, update tables, delete tables, etc.
Once the tables are created, commands can be used to manipulate data into those tables.
The most commonly used command for this purpose is the select command. The select command allows the user to do the following:Specify what information is to be retrieved and from
which tables.Specify conditions to retrieve the data in a specific form.Specify how the retrieved data are to be displayed.
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques4. Integers and Induction5. Relations and Posets
6.Functions7. Counting Principles
8. Boolean Algebra
CSE 2353 f11
Learning Objectives
Learn about functions
Explore various properties of functions
Learn about binary operations
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Functions
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FunctionsEvery function is a relationTherefore, functions on
finite sets can be described by arrow diagrams. In the case of functions, the arrow diagram may be drawn slightly differently.
If f : A → B is a function from a finite set A into a finite set B, then in the arrow diagram, the elements of A are enclosed in ellipses rather than individual boxes.
CSE 2353 f11
Functions
To determine from its arrow diagram whether a relation f from a set A into a set B is a function, two things are checked: 1) Check to see if there is an arrow from each
element of A to an element of B This would ensure that the domain of f is the set
A, i.e., D(f) = A
2) Check to see that there is only one arrow from each element of A to an element of B This would ensure that f is well defined
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Functions
Let A = {1,2,3,4} and B = {a, b, c , d} be sets
The arrow diagram in Figure 5.6 represents the relation f from A into B
Every element of A has some image in B
An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b
CSE 2353 f11
Functions
Therefore, f is a function from A into B
The image of f is the set Im(f) = {a, b, d}
There is an arrow originating from each element of A to an element of B D(f) = A
There is only one arrow from each element of A to an element of B f is well defined
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Functions
The arrow diagram in Figure 5.7 represents the relation g from A into B
Every element of A has some image in B D(g ) = A
For each a ∈ A, there exists a unique element b ∈ B such that g(a) = b g is a function from A into
B
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Functions
The image of g is Im(g) = {a, b, c , d} = B
There is only one arrow from each element of A to an element of B g is well defined
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Functions
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Functions
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Functions Let A = {1,2,3,4} and B = {a, b,
c , d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10
The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it. If a1, a2 ∈ A and a1 = a2, then
f(a1) = f(a2). Hence, f is one-one.
Each element of B has an arrow coming to it. That is, each element of B has a preimage. Im(f) = B. Hence, f is onto B. It
also follows that f is a one-to-one correspondence.
Example 5.1.16
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FunctionsLet A = {1,2,3,4} and
B = {a, b, c , d, e} f : 1 → a, 2 → a, 3 → a,
4 → aFor this function the
images of distinct elements of the domain are not distinct. For example 1 2, but f(1) = a = f(2) .
Im(f) = {a} B. Hence, f is neither one-one nor onto B.
Example 5.1.18
CSE 2353 f11
FunctionsLet A = {1,2,3,4} and
B = {a, b, c , d, e}
f : 1 → a, 2 → b, 3 → d, 4 → e
f is one-one. In this function, for the element c of B, the codomain, there is no element x in the domain such that f(x) = c ; i.e., c has no preimage. Hence, f is not onto B.
CSE 2353 f11
Functions
CSE 2353 f11
Functions
Let A = {1,2,3,4}, B = {a, b, c , d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14.
The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C.
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a Set
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Binary Operations
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CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques4. Integers and Induction5. Relations and Posets
6. Functions
7.Counting Principles8. Boolean Algebra
CSE 2353 f11
Learning Objectives
Learn the basic counting principles—multiplication and addition
Explore the pigeonhole principle
Learn about permutations
Learn about combinations
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Basic Counting Principles
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Basic Counting Principles
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Pigeonhole Principle
The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.
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Pigeonhole Principle
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Permutations
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Permutations
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Combinations
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Combinations
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Generalized Permutations and Combinations
CSE 2353 OUTLINE
1. Sets 2. Logic
3. Proof Techniques4. Integers and Induction5. Relations and Posets
6. Functions7. Counting Principles
8.Boolean Algebra
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Two-Element Boolean AlgebraLet B = {0, 1}.
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Two-Element Boolean Algebra
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Two-Element Boolean Algebra
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Boolean Algebra
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Boolean Algebra
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression.
CSE 2353 f11
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CSE 2353 f11