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Published byPriscilla Pearson Modified over 9 years ago
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snick snack Supplement: Worked Set Proofs Based on work by Meghan Allen
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Proofs with Sets #1 version 1 Prove that for all sets A and B: A B = A B Proof #1: A B = {x U | x ( A B)} = {x U | ~(x A x B)} = {x U | x A x B} = {x U | x A x B} = A B Def’n of De Morgan’s Def’n of Def’n of Def’n of
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Proofs with Sets #1 version 2 Remember that for any two sets C and D, C = D iff C is a subset of D and D is a subset of C. C = D [C D D C]
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Proofs with Sets #1 version 2 a)Prove that: A B A B Pick an arbitrary x A B, Then x A B. ~(x A x B) x A x B x A x B x ( A B) Def’n of De Morgan’s Def’n of Def’n of
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Proofs with Sets #1 version 2 b)Prove that: A B A B Pick an arbitrary x A B Then, x A x B x A x B ~(x A x B) x A B x A B
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Proofs with Sets #1 version 2 conclusion We have shown that A B A B and A B A B. Therefore, A B = A B.
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Proof with Sets #2 Prove that, for all sets A, B and C: (A B) – C = (A – C) (B – C) This time we’ll use the Set Identities LHS: (A B) – C = (A B) C Set Difference Law = C (A B) Commutative Law = (C A) (C B) Distributive Law = (A C) (B C) Commutative Law = (A - C) (B – C) Set Difference Law
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Proofs with Sets #3 Prove or disprove, for all sets A and B, A – B = B – A We can disprove this with a counterexample. Let A = {1,2} and B = {2,3} Then, A-B = {1} and B-A = {3} A – B ≠ B – A
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Proofs with Sets #4 Prove that for all sets A and B, if A B then A B = Proof by contradiction. Assume A B and A B . Then, there exists an element x A B. By the definition of intersection, that means x A and x B. Since x B, x B by the definition of complement. But, A B, so since x A, x B by definition of subset. Thus, x B and x B which is a contradiction. Hence the assumption is false and therefore for all sets A and B, if A B then A B =
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Proofs with Sets #5 Prove that for all sets A and B, if A B then P(A) P(B) WLOG, let A and B be sets. Proof by antecedent assumption: Assume A B. Now, consider X P(A). X A by the definition of power set. But, because A B, X B by the transitive property of subsets* and thus, by the definition of power set, X P(B). This proves that for all X, if X P(A) then X P(B) and so P(A) P(B). *Epp -Theorem 5.2.1
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Proofs with Sets #6 version 1 Prove that for all sets S, S. We proceed by contradiction. Assume that there is a set S such that the empty set is not a subset of S. Then, by definition of subset, there must be some element x of the empty set that is not an element of S. However, the empty set has no elements. This is a contradiction. QED
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Proofs with Sets #6 version 2 Prove that for all sets S, S. Note that S x U, x x S x U, x x S. We proceed by proving this last statement. We know nothing is an element of the empty set: x U, x . By generalization, it follows that: x U, x x S. QED
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