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Review of Inductive Construction of Sets and Inductive Proof Techniques

Review of Inductive Construction of Sets and Inductive Proof Techniques. Example 1.1, page 5: Power Set Construction. Let S = {a, b, c}. Power Set The set of all the subsets of a given set S, is called power set of S. The power set of S is denoted by p (S) or 2 |S| . p (S) = {  }.

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Review of Inductive Construction of Sets and Inductive Proof Techniques

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  1. Review of Inductive Construction of Sets and Inductive Proof Techniques Example 1.1, page 5: Power Set Construction Let S = {a, b, c}. Power Set The set of all the subsets of a given set S, is called power set of S. The power set of S is denoted by p(S) or 2|S|. p(S) = {} a {, {a}} b { , {a}, {b}, {a, b}} C { , {a}, {b}, {c}, {a, b},{a, c}, {b, c}, {a, b,c}} a {, {a}} b { , {a}, {b}, {a, b}}

  2. I II |p(S)| = 2n III |p(S)| = 2n+1 Mathematical Induction Use Mathematical Induction on the size of S to show that if S is a finite set, then Basis: It is trivial to see that if S =  then |S| = 0 and Inductive hypothesis: Assume that |S| = n and The number of elements in the power set of S = 2|S| Inductive step: Let |S| = n+1; assume II is true and show that If we add 1 element to S in II, the size of the power set of S is doubled. The proof is complete.

  3. III Basis: It is clear that n = 1, Sn = 1. I Induction hypothesis: Assume that for some value of n > 1, Induction Step: Using I and II to show that II Inductive Proofs – Example 2 Prove that Example 1.6, page 11/12 Proof by mathematical induction: Denote the sum by Sn.

  4. Add n + 1 to both sides of II to obtain Another proof: , which is the same as III. Adding these two, we obtain, Inductive Proofs – Continued

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