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Discrete Mathematics

Ch2.2 Set Operations

Dr. Talal Bonny

March 13, 2014

Set operations: Union Formal definition for the union of two sets:

A U B = { x | x ∈ A or x ∈ B }

Further examples o {1, 2, 3} U {3, 4, 5} =

{1, 2, 3, 4, 5} o {a, b} U {3, 4} =

{a, b, 3, 4} o {1, 2} U ∅ =

{1, 2} Properties of the union operation

o A U ∅ = A Identity law o A U U = U Domination law o A U A = A Idempotent law o A U B = B U A Commutative law o A U (B U C) = (A U B) U C Associative law 2

Set operations: Intersection Formal definition for the intersection of two sets: A ∩ B = { x | x ∈ A and x ∈ B }

Examples

o {1, 2, 3} ∩ {3, 4, 5} = {3}

o {a, b} ∩ {3, 4} = ∅

o {1, 2} ∩ ∅ = ∅

Properties of the intersection operation o A ∩ U = A Identity law o A ∩ ∅ = ∅ Domination law o A ∩ A = A Idempotent law o A ∩ B = B ∩ A Commutative law o A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law 3

Disjoint sets Formal definition for disjoint sets: two sets are disjoint if

their intersection is the empty set Further examples

o {1, 2, 3} and {3, 4, 5} are not disjoint

o {a, b} and {3, 4} are disjoint

o {1, 2} and ∅ are disjoint Their intersection is the empty set

o ∅ and ∅ are disjoint! Their intersection is the empty set

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Set operations: Difference Formal definition for the difference of two sets:

A - B = { x | x ∈ A and x ∉ B } Further examples

o {1, 2, 3} - {3, 4, 5} = {1, 2}

o {a, b} - {3, 4} = {a, b}

o {1, 2} - ∅ = {1, 2} The difference of any set S with the empty set will be the set S

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Complement sets Formal definition for the complement of a set: A = { x | x ∉ A } = Ac

o Or U – A, where U is the universal set

Further examples (assuming U = Z)

o {1, 2, 3}c = { …, -2, -1, 0, 4, 5, 6, … }

Properties of complement sets

o (Ac)c = A Complementation law o A U Ac = U Complement law o A ∩ Ac = ∅ Complement law

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Set Identities (equivalence) Set identities is a term used to show equality of two

sets. It is similar to the logical equivalences discussed in

Section 1.2. The table in the next slide lists the most important set

identities. The set identities given can be proved directly from the

corresponding logical equivalences.

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Set identities

A∪∅ = A A∩U = A

Identity Law A∪U = U A∩∅ = ∅

Domination law

A∪A = A A∩A = A

Idempotent Law (Ac)c = A Complement

Law

A∪B = B∪A A∩B = B∩A

Commutative Law

(A∪B)c = Ac∩Bc (A∩B)c = Ac∪Bc

De Morgan’s Law

A∪(B∪C) = (A∪B)∪C A∩(B∩C)

= (A∩B)∩C

Associative Law

A∩(B∪C) = (A∩B)∪(A∩C) A∪(B∩C) =

(A∪B)∩(A∪C)

Distributive Law

A∪(A∩B) = A A∩(A∪B) = A

Absorption Law

A ∪ Ac = U A ∩ Ac = ∅

Complement Law

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Example Let A, B, and C be sets. Show that

Solution: We have

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Membership Tables Set identities can also be proved using membership

tables. Consider each combination of sets that an element can

belong to To indicate that an element is in a set, a 1 is used; To indicate that an element is not in a set, a 0 is used. Notice the similarity between membership tables and

truth tables.

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Membership Tables (cont.)

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Computer Representation of Sets One way is to use bitmap First, specify an arbitrary ordering of the elements of U,

for instance . Represent a subset A of U with the bit string of length n,

where the ith bit in this string is 1 if ai belongs to A and is 0 if ai does not belong to A.

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Example 1 U is the set of all students in the class. U = {a,b,c,d,e,f,g,h,i,j,k,l,m} We need to represent

o The students who are in the basketball team {c,d,f,k,m}. Basketball team B = {0011010000101}

o Honor students who have high CGPA {a,d,j,k,m}. Honor list H = {1001000001101}

Basketball team won the tournament and department is doing a party for which both the basketball players and honor students will be invited o B U H

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0011010000101 1001000001101 ------------------- 1011010001101

What if we only invite those members in the team who are also in

the honor list o B ∩ H

0011010000101 1001000001101 ------------------- 0001000000101

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{c,d,f,k,m} {a,d,j,k,m}

Example 1- cont’d

Example 2

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Open Book Problem 2 Write a program, in your preferred language, that takes

two finite sets and lists all elements in the Cartesian product of these two sets. o For simplicity, you can assume all elements are integers

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