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# hw02 - In the Name of God Fall 2005 Discrete Mathematics...

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In the Name of God Fall 2005 Discrete Mathematics Homework 2 Due: Aban 1 Problem 1. Suppose S ( n ) is a predicate on natural numbers, n , and suppose k N S ( k ) S ( k + 2) . Indicate whether each of the following statements are true or false. (a) n m > n [ S (2 n ) ∧ ¬ S (2 m )]. (b) [ n S (2 n )] → ∀ n S (2 n + 2). (c) [ n S ( n )] → ∀ n m > n S ( m ). Problem 2. Let P ( n ) be the statement, ”There exists a decreasing length- n sequence of natural numbers.” We prove that P ( n ) holds for all n by induction. The base case P (1), is clear, as we can just take the single-element sequence { 1 } . For the inductive state, suppose that P ( n ) holds, and let { a 1 , a 2 , · · · , a n } be a corresponding decreasing sequence. Then choose some a 0 larger than max a i , and note that { a 0 , a 1 , a 2 , · · · , a n } forms a length-( n + 1) decreasing sequence. This demonstrates P ( n + 1) and hence completes the proof. What is wrong with this proof? Problem 3. Boolean logic comes up in digital circuit design using the convention that T corresponds to 1 and F to 0.

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