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

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Unformatted text preview: 21-127, Concepts of Mathematics, Review Problems 1. Prove the following using either weak or strong induction: (a) 2n ≥ n + 1. φn −φn + − where F (n) φ+ −φ√ − 1+ 5 and φ+ = 2 . n n 3 2 k=1 k = ( k=1 k ) . n−1 3 n4 k=1 k < 4 . (b) F (n) = (c) (d) = F (n − 1) + F (n − 2), F (0) = 0, F (1) = 1, φ− = √ 1− 5 2 (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 x2 − 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 ∈ N 7. Show that if m divides n then F (m) divides F (n). (Hint: use the result from (9f.) and induction) 8. A function f : A → B is said to be epimorphic if given any set X and any two other functions g, h : B → X , ∀x ∈ A g ◦ f (x) = h ◦ f (x) =⇒ h = g . Prove a function is epimorphic if and only if it is surjective. 9. A function f : A → B is said to be monomorphic if given any set X and any two other functions g, h : X → A, ∀x ∈ X f ◦ g (x) = f ◦ h(x) =⇒ h = g . Prove a function is monomorphic if and only if it is injective. 1 ...
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This note was uploaded on 09/08/2011 for the course MATH 21-127 taught by Professor Gheorghiciuc during the Spring '07 term at Carnegie Mellon.

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