23+PS4 - Handout #23 April 18, 2011 CS103 Robert Plummer...

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Handout #23 CS103 April 18, 2011 Robert Plummer Problem Set #4—Due Monday, April 25 in class 1. Prove by induction that for any natural number n, 3 ) 2 )( 1 ( ) ) 1 ( ( 1 n n n i i n i 2. Let x be an integer. Show that for every integer n 2, x n is even if and only if x is even. Proving the biconditional will naturally involve two subproofs. You are required to use induction in at least one of them, but otherwise, do not use induction where a very simple non-inductive proof will suffice. 3. Prove by induction that for every integer n 2, the complement of the union of any n sets equals the intersection of the complements of these sets. Using set notation, we could state this as: If A 1 , A 2 , . .., A n are sets and n 2, then (A 1 A 2 ... A n )' = A 1 ' A 2 ' ... A n ' . Or, using an overbar rather than "prime" to mean complement, the theorem is: If A 1 , A 2 , . .., A n are sets and n 2, then A 1 A 2 ... A n = A
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This note was uploaded on 06/01/2011 for the course EE 103 taught by Professor Plummer during the Spring '11 term at Stanford.

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23+PS4 - Handout #23 April 18, 2011 CS103 Robert Plummer...

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