Set Operations

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Set Operations

  1. 1. D i s c re teS tru c tu re s (Discrete Mathematics) Topic: Set Operations ©bilalAmjad bilalamjad78633@yahoo.com
  2. 2. Set operations: Union Formal definition for the union of two sets: A U B = { x | x ∈ A or x ∈ B } or A 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
  3. 3. 05/26/12
  4. 4. Set operations: Intersection Formal 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
  5. 5. Exercise-intersection05/26/12
  6. 6. Exercise-union05/26/12
  7. 7. Disjoint sets Formal 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
  8. 8. Set operations: Difference Formal 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
  9. 9. Complement sets Formal definition for the complement of a set: A = { x | x ∉ A } = Ac  Or U – A, where U is the universal set Further 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
  10. 10. Set identities A∪∅ = A A∪U = U Identity Law Domination law A∩U = A A∩∅ = ∅ A∪A = A Idempotent Complementation (Ac)c = A A∩A = A Law LawA∪B = B∪A Commutative (A∪B)c = Ac∩Bc De Morgan’s LawA∩B = B∩A Law (A∩B)c = Ac∪BcA∪(B∪C) A∩(B∪C) == (A∪B)∪C Associative (A∩B)∪(A∩C) Distributive LawA∩(B∩C) Law A∪(B∩C) == (A∩B)∩C (A∪B)∩(A∪C)A∪(A∩B) = A Absorption A ∪ Ac = U Complement LawA∩(A∪B) = Law A ∩ Ac = ∅ A
  11. 11. 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
  12. 12. What we are going to prove… A∩B=B-(B-A) A B B-(B-A) B-A A∩B
  13. 13. 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
  14. 14. Showing each is a subset of the others (A ∩ B)c = Ac ∪ BcProof: 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 ∈ B c ⇒ x ∈ Ac ∪ B c(ii) Similarly we show that Ac ∪ Bc ⊆ (A ∩ B)c
  15. 15. Exercise

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