74hw3 - n(1 2 n 2 = 1 3 2 3 n 3 5(a Find a formula for n X...

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Homework Assignment 3 Due: Wednesday, September 27 1. Let f : X Y . (a) Show that f is surjective if and only if f ( f - 1 [ B ] ) = B for all subsets B Y . (b) Show that f is injective if and only if f - 1 [ f ( A )] = A for all subsets A X . 2. Provide a proof or a counterexample to each of the following. (a) If f : X Y has the property that f ( f - 1 [ f ( A )] ) = f ( A ) for all subsets A X , then f is bijective. (b) If f : X Y has the property that f ( f - 1 f ( f - 1 [ B ] ) ) = B, for all subsets B Y , then f is bijective. 3. Provide a proof or a counterexample to each of the following. (a) A function f : X Y is injective if and only if, for all subsets A, B X , f ( A B ) = f ( A ) f ( B ) . (b) A function f : X Y is surjective if and only if, for all subsets C, D Y , f - 1 [ C D ] = f - 1 [ C ] f - 1 [ D ] . 4. (a) Prove by induction that, for every positive integer n , n k =1 k 3 = n 2 ( n + 1) 2 4 . (b) Prove that, for every positive integer
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Unformatted text preview: n , (1 + 2 + ... + n ) 2 = 1 3 + 2 3 + ... + n 3 . 5. (a) Find a formula for n X k =1 1 2 k , and prove your answer by induction. (b) Find a formula for n X k =1 1 3 k , and prove your answer by induction. 6. Find a formula for the sum of the first n fourth powers, and prove your answer by induction. Also explain how you were able to guess the right formula. 7. Give a proof or a counterexample for each. (a) Suppose that f : X → Y and g : X → Y . Then for all subsets B ⊆ Y , f ( g-1 [ B ]) = g ( f-1 [ B ]) . (b) Suppose that f : X → Y and g : X → Y . Then for all subsets A ⊆ X , f-1 [ g ( A )] = g-1 [ f ( A )] ....
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