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Properties of Sets

Properties of Sets. Lecture 26 Section 5.2 Tue, Mar 6, 2007. Proving Basic Properties. Theorem: Let A and B be sets. Then A  B  A . Proof: Let x  A  B . Then x  A and x  B . Therefore, x  A . So A  B  A . Comments.

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Properties of Sets

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  1. Properties of Sets Lecture 26 Section 5.2 Tue, Mar 6, 2007

  2. Proving Basic Properties • Theorem: Let A and B be sets. Then AB A. • Proof: • Let x AB. • Then x Aandx B. • Therefore, x A. • So AB A.

  3. Comments • The proof uses the logic that S T if and only if x S x T. • A Venn diagram alone does not constitute a proof. • This theorem is suggestive of the “specialization” argument form (p. 40) p q p • What is the connection?

  4. Identity Laws • Theorem: Let A be any set. Then A U = A, A   = A.

  5. DeMorgan’s Laws • Theorem: Let A and B be any two sets. Then (A B)c = Ac  Bc, (AB)c = Ac  Bc.

  6. Set-Theoretic Proofs • Theorem: Let A, B, and C be sets. Then (A – B)  (C – B) = (AC) – B. • Proof: • Let x (A – B)  (C – B). • Then x A – Band x C – B, • x A, and x C, and x B, •  xAC and x B, • x (AC) – B.

  7. Set-Theoretic Proof • It then follows that (A – B)  (C – B)  (AC) – B. • The second half of the proof will show that (AC) – B (A – B)  (C – B). • However, the logic is exactly the reverse of the first half. • We may handle that by saying “and conversely” at the end of the first half.

  8. Comment • The preceding theorem is equivalent to the logical equivalence (pq)  (r q) = (p  r)  q which is not hard (at all!) to prove. • Proof: • (pq)  (r q) = pq  r q = p r q = (p  r)  q.

  9. Question • Why was that so much easier than the original proof? • Because we know a lot about the operators , , and . • We could use the “algebra” of , , and . • Is there an algebra of , , and complement?

  10. Algebraic Properties of Sets • See Theorem 1.1.1, p. 14. • Commutative Laws: • A B = B  A. • A B = B  A. • Associative Laws: • (A  B)  C = A  (B  C). • (A  B)  C = A  (B  C).

  11. Algebraic Properties of Sets • Distributive Laws: • A (BC) = (AB)  (AC). • A (BC) = (AB)  (AC). • Identity Laws: • AU = A. • A = A.

  12. Algebraic Properties of Sets • Complement Laws: • AAc = U. • AAc = . • Double Complement Law: • (Ac)c = A. • Idempotent Laws: • AA = A. • AA = A.

  13. Algebraic Properties of Sets • Universal Bound Laws: • AU = U. • A = . • DeMorgan’s Laws: • (AB)c = AcBc. • (AB)c = AcBc.

  14. Algebraic Properties of Sets • Absorption Laws: • A (AB) = A. • A (AB) = A. • Complements of U and . • Uc = . • c = U.

  15. The Proof Revisited • Theorem: Let A, B, and C be sets. Then (A – B)  (C – B) = (AC) – B. • Proof: • (A – B)  (C – B) = (A Bc)  (C  Bc) = (A  C)  (Bc  Bc) = (A  C)  Bc = (A  C) – B.

  16. The Proof Revisited • Theorem: Let A, B, and C be sets. Then (A – B)  (C – B) = (AC) – B. • Proof:

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