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Set Theory

Second Part

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Disjoint Set

• let A and B be a set.• the two sets are called disjoint if their

intersection is an empty set.• Intersection of set:

A={1,3,5,7,9} B={2,4,6,8,10}

• A B= { } = , thus A and B are disjoint

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Cartesian Product • Ordered n-tuples has a1 as its first element, a2 as its

second element, …and an as its nth element.

• (a1,a2,…,an). • 2 ordered n-tuples are equal if and only if each

corresponding pair of their elements is equal. (a1,a2,…,an) = (b1,b2,…,bn) if & only if ai = bi for i = 1,2,…,n.

• 2-tuples are called ordered pairs: ordered pairs (a,b) & (c,d) are equal if a = c & b = d. ordered pairs (a,b) & (b,a) are equal if a = b.

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Cartesian Product

• Is (1,2) = (2,1)? No.• Is (3,(-2)2,1/2) = (√9,4,3/6)? Yes.• Let A and B be sets. The Cartesian

product of A and B, denoted by A x B, is the set of all ordered pairs (a, b) where

• a A b B.• Hence, A x B = { (a,b) | a A b B}.

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Cartesian Product• E.g. • What is the Cartesian product of A = {1,2} and

B = (a,b,c} ?• A x B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}.• The Cartesian product of the set A1, A2, … , An,

denoted by A1 x A2 x … x An is the set of ordered n-tupples (a1, a2,…,an), where ai belongs to Ai for i=1,2,…,n.

• In other words,• A1 x A2 x … x An = {(a1, a2,

…,an) | ai Ai for i =1,2,…,n}

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Set Theory

ObjectivesOn completion of this chapter, student

should be able to:• Identify element and subset of a set• Perform set operations• Establish set properties (equalities) using

element arguments.

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Properties of Set• Subset relations1. Inclusion of Intersection: For all sets A and B, (a) (AB) A and (b) (A B) B2. Inclusion in Union (a) A AB and (b) B AB3. Transitive of Subsets if A B and B C, then A C(One set is a subset of another)

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Element Argument

• The basic method for proving that one set is a subset of another.

Let sets X and Y be given. To prove that X Y 1. Suppose that x is a particular but

arbitrarily chosen element of x2. Show that x is an element of Y.

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Example

• Prove that for all sets A and B, AB A• Proof:• Suppose x is any element of AB.• Then xA and xB by definition of

intersection. In particular xA.

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Set Identities

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Set Identities

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Set Identities

11. Uc= and c=U Complements of U & 12. A – B = A Bc Set Difference Law

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Proving Set Identities

• Basic Method• Let sets X and Y be given. To prove that X = Y:1. Prove that X Y2. Prove that Y X.

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Distributive Law• Prove that A(B C) = (A B) (A C)Suppose that x A(B C).Then x A or x (B C).Case 1: (x A) Since xA then xAB and also xAC by

definition of unionHence x(A B) (A C)Case 2: (xBC)Since (xBC), then xB and x C. Since xB,

xA B and since x C, x A C. Hence x(A B) (A C)

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• Show that (A B) (A C) A(B C) Suppose that x (A B) (A C) Then x (A B) and x (A C).Case 1: (x A) Since xA we can conclude that A(B C) Case 2: (x A )Since x A, then xBC. It follows thatx A(B C). Hence (A B)(AC) A(BC) Therefore, A(B C) = (A B) (A C)

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Question

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De Morgan’s Law• Prove that (AB)c = AcBc

• Show that (AB)c AcBc

• x (AB)c then x (AB).• It means x A and x B by DeMorgan’s Law.

Hence xAc and xBc or x AcBc

• So (AB)c AcBc

• Next is to show that• AcBc (AB)c

• Do it yourself!

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Algebraic Proofs of Set Identities

• Can use set identities to derive new set identities or to simplify a complicated set expression AB)c AcBc

• Prove that (A B)–C = (A–C) (B–C) (A B)–C = (A B) Cc by the set difference law = Cc (A B) by the comm. law = (Cc A)(CcB) by the distributive law = (ACc)(BCc) comm. law = (A-C)(B-C) by the set difference law

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Question

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Another example

• Prove that A-(AB)=(A-B) A-(AB)= A(AB)c by the set difference law = A (Ac Bc) by the De Morgan’s law = (A Ac)(ABc) the distributive law = (ABc) the complement law = (ABc) the identity law = A-B by the set difference law

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Activity

• Use set identities to:• Show that (A (B C))c = (Cc Bc) Ac

• Simplify the expression• [((A B) C)c Bc)]c

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Summary• Covered:• LO: Identify element and subset of a set Basic definitions: e.g. element and subset of a set,

empty set, power set, cartesian product• LO: Perform set operations Union, intesection, set difference, complement,

symmetric difference.• LO: Establish set properties (equalities) using element

arguments. Proving set identities using element argument

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TERIMA KASIH