Discrete Structures

(Discrete Mathematics)

Topic: Set Operations

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Set operations: UnionFormal definition for the union of two sets:

A U B = { x | x A or x B } orA U B = { x U| x A or x B }

Further examples {1, 2, 3} {3, 4, 5} = {1, 2, 3, 4, 5} {a, b} {3, 4} = {a, b, 3, 4} {1, 2} = {1, 2}

Properties of the union operation A = A Identity law A U = U Domination law A A = A Idempotent law A B = B A Commutative law A (B C) = (A B) C Associative law

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Set operations: IntersectionFormal definition for the intersection of two sets:

A ∩ B = { x | x A and x B }Examples

{1, 2, 3} ∩ {3, 4, 5} = {3} {a, b} ∩ {3, 4} = {1, 2} ∩ =

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

Exercise-intersection

04/12/23

Exercise-union

04/12/23

Disjoint setsFormal definition for disjoint sets:

two sets are disjoint if their intersection is the empty set

Further examples {1, 2, 3} and {3, 4, 5} are not disjoint {a, b} and {3, 4} are disjoint {1, 2} and are disjoint

• Their intersection is the empty set and are disjoint!

• Because their intersection is the empty set

Set operations: DifferenceFormal definition for the difference of two sets:

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

{1, 2, 3} - {3, 4, 5} = {1, 2} {a, b} - {3, 4} = {a, b} {1, 2} - = {1, 2}

• The difference of any set S with the empty set will be the set S

Complement sets

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

Or U – A, where U is the universal setFurther examples (assuming U = Z)

{1, 2, 3}c = { …, -2, -1, 0, 4, 5, 6, … } {a, b}c = Z

Properties of complement sets (Ac)c = A Complementation law A Ac = U Complement law A ∩ Ac = Complement law

Set identitiesA = AAU = A

Identity LawAU = UA =

Domination law

AA = AAA = A

Idempotent Law

(Ac)c = AComplementation

Law

AB = BAAB = BA

Commutative Law

(AB)c = AcBc

(AB)c = AcBcDe Morgan’s Law

A(BC) = (AB)CA(BC)

= (AB)C

Associative Law

A(BC) = (AB)(AC)A(BC) =

(AB)(AC)

Distributive Law

A(AB) = A

A(AB) = A

Absorption Law

A Ac = UA Ac =

Complement Law

How to prove a set identity

For example: A∩B=B-(B-A)Four methods:

Use the basic set identities Use membership tables Prove each set is a subset of each other Use set builder notation and logical equivalences

What we are going to prove…A∩B=B-(B-A)

A B

A∩BB-A

B-(B-A)

Proof by Set Identities

A B = A - (A - B) = B – (B – A)

Proof: A - (A - B) = A - (A Bc) = A (A Bc)c

= A (Ac B) = (A Ac) (A B) = (A B) = A B

Showing each is a subset of the others

(A B)c = Ac Bc

Proof: Want to prove that (A B)c Ac Bc and Ac Bc (A B)c (i) x (A B)c

x (A B) (x A B) (x A x B) (x A) (x B) x A x B x Ac x Bc

x Ac Bc

(ii) Similarly we show that Ac Bc (A B)c

Exercise