09A Week 9 - CMPUT 272(Stewart Lecture 15 Reading Epp 1.2...

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CMPUT 272 (Stewart) Lecture 15 Reading: Epp 1.2, 1.3, 6, 7.1, 7.2 Sets (continued) Recall: Proofs in set theory: Venn diagram may help to: determine whether a claim is true or false find a counterexample to a false universal claim element method used to: prove that one set is a subset of another prove set identities prove that a set is equal to (by contradiction) known (famous) set identities: simplify set expressions give algebraic proofs of other set identities the famous set identities can be proven by the element method mathematical induction 1
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Theorem. For all n N , if a set X has n elements, then |P ( X ) | = 2 n . Proof (by mathematical induction): Let P ( n ) mean: any set with n elements has 2 n subsets. Basis step: The only set with zero elements is the empty set, which has 2 0 = 1 subset. Therefore P (0) is true. Inductive step: Let k N such that any set with k elements has 2 k subsets. [We must show that any set with k + 1 elements has 2 k +1 subsets.] Let X be a set with k + 1 elements. Since k + 1 > 0, X has at least one element. Let z X . Now each subset of X either contains z or not. Each subset of X that does not contain z is a subset of X - { z } and each subset of X - { z } is a subset of X that does not contain z . So the number of subsets of X that do not contain z is equal to the number of subsets of X - { z } . Each subset of X that contains z is equal to the union of { z } and a subset of X - { z } , and the union of { z } and any subset of X - { z } is a subset of X that contains z . So the number of subsets of X that contain z is equal to the number of subsets of X - { z } .
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