Discrete Mathematics

Discrete Mathematics

It is assumed that you have studied basics of mathematics before. • The slides in this section are for your review. They will not all be covered in class. • If you need extra help in this area, a special help session will be scheduled. • A small test based on multiple choice questions will help you to analyse yourself

Learning Objectives • Introduction • Sets • Logic & Boolean Algebra • Proof Techniques • Counting Principles • Relations and Functions • Partial Ordered Sets • Group Theory • Graphs and Trees

Introduction

Introduction • What it isn’t: continuous • Discrete: consisting of distinct or unconnected elements • Countably Infinite • DefinitionDiscrete Mathematics is a collection of mathematical topics that examine and use finite or countably infinite mathematical objects.

Sets

Sets • 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 ver model involves requests and replies.

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}

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+ : R : The set of all real numbers

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

Sets • Superset • 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

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) 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

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 setwith 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

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

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

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

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

Intersection Of Sets • Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, • then X ∩ Y = {5}

Disjoint sets • Example: If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y = 

Difference of Sets • Example: If X = {a,b,c,d} and Y = {c,d,e,f}, then X – Y = {a,b} and Y – X = {e,f}

Compliment • Compliment of a Set • The complement of a set X with respect to a universal set U, denoted by , is defined to be • = {x |x  U, but x  X} • Example: If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then ~X = {a,b}

Sets • Ordered 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 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)}

Logic & Boolean Algebra

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

Mathematical Logic • A statement, or a proposition, is a declarative sentence that is either true or false, but not both • Uppercase letters denote propositions • Examples: • P: 2 is an even number (true) • Q: 7 is an even number (false) • R: A is a vowel (true) • The following are not propositions: • P: My cat is beautiful • Q: My house is big

Mathematical Logic • 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

Mathematical Logic • 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

Mathematical Logic • Disjunction • Let 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 or q is true; otherwise p v q is false • The symbol v is read “or”

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

Mathematical Logic • Implication • Truth Table for Implication: • p is called the hypothesis, q is called the conclusion

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

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”

Mathematical Logic • Biconditional • Truth Table for the Biconditional:

Boolean Algebra • Boolean algebra provides the operations and the rules for working with the set {0, 1}. • We are going to focus on three operations: • Boolean complementation, • Boolean sum, and • Boolean product

Boolean Algebra • The complementis denoted by a bar (on the slides, we will use a minus sign). It is defined by • -0 = 1 and -1 = 0. • The Boolean sum, denoted by + or by OR, has the following values: • 1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0 • The Boolean product, denoted by  or by AND, has the following values: • 1  1 = 1, 1  0 = 0, 0  1 = 0, 0  0 = 0

Boolean Algebra • Definition: Let B = {0, 1}. The variable x is called a Boolean variable if it assumes values only from B. • A function from Bn, the set {(x1, x2, …, xn) |xiB, 1  i  n}, to B is called a Boolean function of degree n. • Boolean functions can be represented using expressions made up from the variables and Boolean operations

Boolean Algebra • The Boolean expressions in the variables x1, x2, …, xn are defined recursively as follows: • 0, 1, x1, x2, …, xn are Boolean expressions. • If E1 and E2 are Boolean expressions, then (-E1), (E1.E2), and (E1 + E2) are Boolean expressions. • Each Boolean expression represents a Boolean function. The values of this function are obtained by substituting 0 and 1 for the variables in the expression.

Boolean Functions and Expressions • For example, we can create Boolean expression in the variables x, y, and z using the “building blocks”0, 1, x, y, and z, and the construction rules: • Since x and y are Boolean expressions, so is xy. • Since z is a Boolean expression, so is (-z). • Since xy and (-z) are expressions, so is xy + (-z).

Boolean Functions and Expressions • Example: Give a Boolean expression for the Boolean function F(x, y) as defined by the following table: Possible solution: F(x, y) = (-x)y

Basic Identities of Boolean Algebra • x + 0 = x • x · 0 = 0 • x + 1 = 1 • x · 1 = 1 (5) x + x = x (6) x · x = x (7) x + x’ = x (8) x · x’ = 0 (9) x + y = y + x (10) xy = yx

Duality Duality Principle – every valid Boolean expression (equality) remains valid if the operators and identity elements are interchanged, as follows: • + . • 1  0 Example: Given the expression • a + (b.c) = (a+b).(a+c) • then its dual expression is • a . (b+c) = (a.b) + (a.c)

Standard Forms Certain types of Boolean expressions lead to gating networks which are desirable from implementation viewpoint. Two Standard Forms: Sum-of-Products and Product-of-Sums Literals: a variable on its own or in its complemented form. Examples: x, x' , y, y' Product Term: a single literal or a logical product (AND) of several literals. • Examples: x, x.y.z', A'.B, A.B, e.g'.w.v

Standard Forms Sum Term: a single literal or a logical sum (OR) of several literals. • Examples: x, x+y+z', A'+B, A+B, c+d+h'+j Sum-of-Products (SOP) Expression: a product term or a logical sum (OR) of several product terms. • Examples: x, x+y.z', x.y'+x'.y.z, A.B+A'.B', A + B'.C + A.C' + C.D Product-of-Sums (POS) Expression: a sum term or a logical product (AND) of several sum terms. • Examples: x, x.(y+z'), (x+y').(x'+y+z), (A+B).(A'+B'), (A+B+C).D'.(B'+D+E')

Proof Techniques

Proof Techniques • Learn various proof techniques • Direct • Indirect • Contradiction • Induction

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

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 Choose Method (Universal Generalization), ∀x (P(x) → Q(x)) is true

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

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.

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 → Pare true • Assume that Pis true and show that Q is true • Assume that Q is true and show that Pis true

Discrete Mathematics