hw01 - In the Name of God Fall 2005 Discrete Mathematics...

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Unformatted text preview: In the Name of God Fall 2005 Discrete Mathematics Homework 1 Due: Mehr 24 Problem 1. Consider the following sequence of predicates: Q 1 ( x 1 ) = x 1 Q 2 ( x 1 , x 2 ) = x 1 ⇒ x 2 Q 3 ( x 1 , x 2 , x 3 ) = ( x 1 ⇒ x 2 ) ⇒ x 3 Q 4 ( x 1 , x 2 , x 3 , x 4 ) = (( x 1 ⇒ x 2 ) ⇒ x 3 ) ⇒ x 4 Q 5 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = ((( x 1 ⇒ x 2 ) ⇒ x 3 ) ⇒ x 4 ) ⇒ x 5 ··· Let T n be the number of different true/false settings of the variables x 1 , x 2 , ··· , x n for which Q n ( x 1 , x 2 , ··· , x n ) is true. For example T 2 = 3 since Q 2 ( x 1 , x 2 ) is true for three different settings of the variables x 1 and x 2 . (a) Express T n +1 in terms of T n , assuming n ≥ 1. (b) Use induction to prove that T n = 1 3 (2 n +1 + (- 1) n ) for n ≥ 1. You may assume your answer to the previous part without proof. Problem 2. Determine whether or not (( A ↔ (( ¬ B ) ∨ C )) → (( ¬ A ) → B )) is a tautology. DO NOT use the truth table....
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hw01 - In the Name of God Fall 2005 Discrete Mathematics...

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