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Exam2Practice - 21-127 Concepts of Mathematics Review...

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21-127, Concepts of Mathematics, Review Problems 1. Prove the following using either weak or strong induction: (a) 2 n n + 1. (b) F ( n ) = φ n + - φ n - φ + - φ - where F ( n ) = F ( n - 1)+ F ( n - 2), F (0) = 0, F (1) = 1, φ - = 1 - 5 2 and φ + = 1+ 5 2 . (c) n k =1 k 3 = ( n k =1 k ) 2 . (d) n - 1 k =1 k 3 < n 4 4 . (e) F ( a + b ) = F ( a ) F ( b - 1) + F ( a + 1) F ( b ) where a 0, b 1 and F is defined above. 2. Prove that f f is injective if and only if f is injective. 3. Prove that f : ( - 1 , 1) R given by f ( x ) = x x 2 - 1 is a bijection. 4. Find an explicit bijection from (0 , 1) to [4 , 9]. 5. Given f : A B and g : B C (a) Prove that if f is surjective and g is not injective, then g f is not injective. (b) Prove that if f is not surjective and g is injective, then g f is not surjective. 6. Prove that n p is irrational for all prime numbers p and n 2
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