Use mathematical induction to prove that for every integer n > 2, if a set S has n elements, then the mumber of subsets of S with an even number of elements equals the number of subsets of S with an odd number of elements. Hint: See the proof of Theorem 6.2.16.

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Q: Let r ER andne N. Prove, by induction on n, that p" -1= (r – 1) · (r"-1 + r"-2 +...+r+1)

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Q: 1. Use induction to prove that Σ m(n+1) for all n eN. k=0

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Q: use generalized induction to prove the given statement. n3 &lt; n ! for all integers n ≥ 6

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Q: 2) Let X be a finite set. Show by induction on n = = [X| that %3D |P(X)| = 2X1.

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Q: 2k >1+ k

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Q: Prove that by mathematically Induction. 2^n < (n + 1)!, for all integers n ≥ 2.

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.