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74-hw5

# 74-hw5 - MATH 74 HOMEWORK 5 DUE MONDAY 3/5 1 For each of...

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MATH 74 HOMEWORK 5: DUE MONDAY 3/5 1. For each of the following functions, tell me: Is the function injective? Is the function surjective? You do not need to prove your answers, or do anything besides answer yes or no for each question. (The point is just to get you thinking about these concepts. If you want to give full proofs, feel free.) (a) f : N N given by f ( x ) = x 2 (b) f : [0 , ) R given by f ( x ) = x 3 (c) f : [ - π/ 2 , π/ 2] [ - 1 , 1] given by f ( x ) = sin x (d) f : P ( N ) → P ( N ) given by f ( A ) = A ∪ { 1 , 2 , 3 } (e) f : P ( N ) → P ( N ) given by f ( A ) = { x N : x A } 2. Let A be a finite set. Let B denote the set of all functions with domain A and codomain { 0 , 1 } . Prove that #( P ( A )) = #( B ). [Hint: come up with a function from P ( A ) B , or vice versa, and prove that it is a bijection.] 3. Definition. If f : X Y is a function, and S X , we let f | S denote the function whose domain is S , whose codomain is Y , and whose rule is the same rule as f . (In other words, f | S is the function S Y you get by thinking of f as a function only on S and not on all of X .) In

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74-hw5 - MATH 74 HOMEWORK 5 DUE MONDAY 3/5 1 For each of...

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