Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications

Learning Objectives Learn about sets Explore various operations on sets Become familiar with Venn diagrams Learn how to represent sets in computer memory Learn about statements (propositions) Discrete Mathematical Structures: Theory and Applications

Learning Objectives Learn how to use logical connectives to combine statements Explore how to draw conclusions using various argument forms Become familiar with quantifiers and predicates Learn various proof techniques Explore what an algorithm is Discrete Mathematical Structures: Theory and Applications

Sets Definition: Well-defined collection of distinct objects Members or Elements: part of the collection Roster Method: Description of a set by listing the elements, enclosed with braces Examples: 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 Discrete Mathematical Structures: Theory and Applications

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} Discrete Mathematical Structures: Theory and Applications

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 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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Sets Superset Proper Subset X 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 Subset X 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 Discrete Mathematical Structures: Theory and Applications

Sets Set Equality Empty (Null) Set 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) Set A Set is Empty (Null) if it contains no elements. The Empty Set is written as  The Empty Set is a subset of every set Discrete Mathematical Structures: Theory and Applications

Sets Finite and Infinite Sets X 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 Discrete Mathematical Structures: Theory and Applications

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 H = { 4 }, |H| = 1, H is a singleton Discrete Mathematical Structures: Theory and Applications

Sets Power Set Universal 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 Discrete Mathematical Structures: Theory and Applications

Sets Venn Diagrams Abstract 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 Discrete Mathematical Structures: Theory and Applications

Sets Union of Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X∪Y = {1,2,3,4,5,6,7,8,9} Discrete Mathematical Structures: Theory and Applications

Sets Intersection of Sets Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5} Discrete Mathematical Structures: Theory and Applications

Sets Disjoint Sets Example: If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y =  Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

Sets The union and intersection of three,four,or even infinitely many sets can be considered For a finite collection of n sets, X1, X2, … Xn where n ≥ 2 : Discrete Mathematical Structures: Theory and Applications

Sets Index Set Discrete Mathematical Structures: Theory and Applications

Sets Example: If A = {a,b,c}, B = {x, y, z} and C = {1,2,3} then A ∩ B =  and B ∩ C =  and A ∩ C = . Therefore, A,B,C are pairwise disjoint Discrete Mathematical Structures: Theory and Applications

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} Discrete Mathematical Structures: Theory and Applications

Sets Complement Example: If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b} Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

Sets Discrete Mathematical Structures: Theory and Applications

Sets Ordered Pair Cartesian Product 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 Product The Cartesian product of two sets X and Y ,written X × Y ,is the set X × Y ={(x,y)|x ∈ X , y ∈ Y} For any set X, X ×  =  =  × X Example: 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)} Discrete Mathematical Structures: Theory and Applications

Sets Diagonal of a Set δx = {(x,x) | x ∈ X} For a set X ,the set δx , is the diagonal of X, defined by δx = {(x,x) | x ∈ X} Example: X = {a,b,c}, δx = {(a,a), (b,b), (c,c)} Discrete Mathematical Structures: Theory and Applications

Sets Computer Representation of Sets A Set may be stored in a computer in an array as an unordered list Problem: Difficult to perform operations on the set. Solution: use Bit Strings A Bit String is a sequence of 0s and 1s Length of a Bit String is the number of digits in the string Elements appear in order in the bit string A 0 indicates an element is absent, a 1 indicates that the element is present Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic A statement, or a proposition, is a declarative sentence that is either true or false, but not both Lowercase letters denote propositions Examples: 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? Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Truth value Negation One of the values “truth” or “falsity” assigned to a statement True is abbreviated to T or 1 False is abbreviated to F or 0 Negation The negation of p, written ∼p, is the statement obtained by negating statement p Truth values of p and ∼p are opposite Symbol ~ is called “not” ~p is read as as “not p” Example: p: A is a consonant ~p: it is the case that A is not a consonant q: Are you in charge? Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Truth Table Conjunction Let 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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Conjunction Truth Table for Conjunction: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Disjunction Let p and q be statements. The disjunction of p and q, written p ∨ q , is the statement formed by joining statements p and q using the word “or” The statement p∨q is true if at least one of the statements p and q is true; otherwise p∨q is false The symbol ∨ is read “or” Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Disjunction Truth Table for Disjunction: Discrete Mathematical Structures: Theory and Applications

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 p  q is read: “If p, then q” “p is sufficient for q” q if p q whenever p Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Implication Truth Table for Implication: p is called the hypothesis, q is called the conclusion Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Implication Let p: Today is Sunday and q: I will wash the car. The conjunction p  q is the statement: p  q : If today is Sunday, then I will wash the car The converse of this implication is written q  p If I wash the car, then today is Sunday The inverse of this implication is ~p  ~q If today is not Sunday, then I will not wash the car The contrapositive of this implication is ~q  ~p If I do not wash the car, then today is not Sunday Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Biimplication Let 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  q is read: “p if and only if q” “p is necessary and sufficient for q” “q if and only if p” “q when and only when p” Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Biconditional Truth Table for the Biconditional: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Statement Formulas Definitions Symbols p ,q ,r ,...,called statement variables Symbols ~, ∧, ∨, →,and ↔ are called logical connectives A statement variable is a statement formula If A and B are statement formulas, then the expressions (~A ), (A ∧ B) , (A ∨ B ), (A → B ) and (A ↔ B ) are statement formulas Expressions are statement formulas that are constructed only by using 1) and 2) above Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Precedence of logical connectives is: ~ highest ∧ second highest ∨ third highest → fourth highest ↔ fifth highest Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Example: Let A be the statement formula (~(p ∨q )) → (q ∧p ) Truth Table for A is: Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Tautology Contradiction 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 Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Logically Implies Logically Equivalent A 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 Equivalent A 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 (or A ⇔ B) Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Proof of (~p ∧q ) → (~(q →p )) Discrete Mathematical Structures: Theory and Applications

Mathematical Logic Proof of (~p ∧q ) → (~(q →p )) [Continued] Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Modus Ponens (Method of Affirming) Modus Tollens (Method of Denying) Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Disjunctive Syllogisms Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Hypothetical Syllogism Dilemma Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Conjunctive Simplification Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Disjunctive Addition Discrete Mathematical Structures: Theory and Applications

Validity of Arguments Valid Argument Forms Conjunctive Addition Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Quantifiers and First Order Logic Predicate or Propositional Function Example: Q(x,y) : x > y, where the Domain is the set of integers Q is a 2-place predicate Q is T for Q(4,3) and Q is F for Q (3,4) Discrete Mathematical Structures: Theory and Applications

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: Discrete Mathematical Structures: Theory and Applications

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 Variable The variable appearing in: or Discrete Mathematical Structures: Theory and Applications

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, Discrete Mathematical Structures: Theory and Applications

Quantifiers and First Order Logic Negation of Predicates (DeMorgan’s Laws) Discrete Mathematical Structures: Theory and Applications

Quantifiers and First Order Logic Formulas in Predicate Logic All statement formulas are considered formulas Each n, n =1,2,...,n-place predicate P( ) containing the variables is a formula. If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A →B and A↔B are statement formulas, where ~, ∧, ∨, → and ↔ are logical connectives If A is a formula and x is a variable, then ∀x A(x) and ∃x A(x) are formulas All formulas constructed using only above rules are considered formulas in predicate logic Discrete Mathematical Structures: Theory and Applications

Quantifiers and First Order Logic Additional Rules of Inference If the statement ∀x P(x) is assumed to be true, then P(a) is also true,where a is an arbitrary member of the domain of the discourse. This rule is called the universal specification (US) If P(a) is true, where a is an arbitrary member of the domain of the discourse, then ∀x P(x) is true. This rule is called the universal generalization (UG) If the statement ∃x P (x) is true, then P(a) is true, for some member of the domain of the discourse. This rule is called the existential specification (ES) If P(a) is true for some member a of the domain of the discourse, then ∃x P(x) is also true. This rule is called the existential generalization (EG) Discrete Mathematical Structures: Theory and Applications

Quantifiers and First Order Logic Counterexample An argument has the form ∀x (P(x ) → Q(x )), where the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication To show that ∀x (P(x) → Q(x)) is false by finding an x in D such that P(x) → Q(x) is false is called the disproof of the given statement by counterexample Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Proof Techniques Direct Proof or Proof by Direct Method Proof 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 true By the rule of Universal Generalization (UG), ∀x (P(x) → Q(x)) is true Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Proof Techniques Proof by Contradiction Assume that the conclusion is not true and then arrive at a contradiction Example: Prove that there are infinitely many prime numbers Proof: 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 Discrete Mathematical Structures: Theory and Applications

Proof Techniques Discrete Mathematical Structures: Theory and Applications

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 true Assume that p is true and show that q is true Assume that q is true and show that p is true Discrete Mathematical Structures: Theory and Applications

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 Or, prove that p if and only if q, and then q if and only if r Other methods are possible Discrete Mathematical Structures: Theory and Applications

Algorithms Definition: step-by-step problem-solving process in which a solution is arrived at in a finite amount of time All algorithms have the following properties: Input : For example, a set of numbers to find the sum of the numbers Output : For example, the sum of the numbers Precision : Each step of the algorithm is precisely defined Uniqueness : Results of each step are unique and depend on the input and results of previous step Finiteness : Algorithm must terminate after executing a finite number of steps Generality : Algorithm is general in that it applies to a set of inputs Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions The symbol := is called the assignment operator Example: The statement x := a is read as “assign the value a to x” or “x gets the value a” or “copy the value of a into x” x := a is also known as an assignment statement Control Structures One way-selection if booleanExpression then statement If booleanExpression evaluates to true, statement is evaluated Two way-selection if booleanExpression then statement1 else statement2 If booleanExpression evaluates to true , statement1 executes, otherwise statement2 executes Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Control Structures The while loop takes the form: while booleanExpression do loopBody The booleanExpression is evaluated. If it evaluates to true, loopBody executes. Thereafter loopBody continues to execute as long as booleanExpression is true The for loop takes the form: for var := start to limit do loopBody var is an integer variable. The variable var is set to the value specified by start. If var limit, loopBody executes. After executing the loopBody , var is incremented by 1. The statement continues to execute until var > limit Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Control Structures The do/while loop takes the form: do loopBody while booleanExpression The loopBody is executed first and then the booleanExpression is evaluated. The loopBody continues to execute as long as the booleanExpression is true Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Block of Statement To consider a set of statements a single statement, the statements are written between the words begin and end begin statement1 statement2 ... statementn; end Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Return Statement The return statement is used to return the value computed by the algorithm and it takes the following form: return expression; The value specified by expression is returned. In an algorithm, the execution of a return statement also terminates the algorithm Read and Print Statements read x; Read the next value and store it in the variable x print x; Output the value of x Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Arrays (List) A list is a set of elements of the same type The length of the list is the number of elements in the list L[1...n ]. L is an array of n components, indexed 1 to n . L[i ] denotes the ith element of L For data in tabular form, a two-dimensional array is used: M[1...m,1...n ] M is a two-dimensional array of m rows and n columns The rows are indexed 1 to m and the columns are indexed 1 to n M[i,j] denotes the (i,j)th element of M, that is, the element at the ith row and jth column position Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Subprograms (Procedures) In a programming language,an algorithm is implemented in the form of a subprogram, a.k.a. a subroutine or a module Two types of subprograms Functions Returns a unique value Procedure Other types of subprograms body of the function or procedure is enclosed between the words begin and end the execution of a return statement in a function terminates the function Discrete Mathematical Structures: Theory and Applications

Algorithms Pseudocode Conventions Comments In describing the steps of an algorithm, comments are included wherever necessary to clarify the steps Two types of comments: single-line and multi-line Single-line comments start anywhere in the line with the pair of symbols // Multi-line comments are enclosed between the pair of symbols /* and */ Specifies what the algorithm does, as well as the input and output Discrete Mathematical Structures: Theory and Applications

Algorithms Polynomial Operations Polynomial in one variable is ai are real (or complex) and n is a non-negative integer If p(x) = a0 then p(x) is a constant polynomial If p(x) is a nonzero constant polynomial, then the degree of p(x) is defined to be 0 If p(x) is not constant and an 0, then n is called the degree of p(x); that is, the degree of a non-constant polynomial is defined to be the exponent of the highest power of x The basic operations performed on polynomials are to add, subtract, multiply, and divide polynomials, and to evaluate a polynomial at a given point Discrete Mathematical Structures: Theory and Applications