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01sol

# 01sol - ASSIGNMENT 1 SOLUTIONS MAT 472 FALL 2011 Problem...

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ASSIGNMENT 1 · SOLUTIONS MAT 472 · FALL 2011 Problem 1 (Exercise 1.2.2) . True or False? If false, provide a counter-example. (a) If A 1 A 2 A 3 · · · are infinite sets, then n =1 A n is infinite. (b) If A 1 A 2 A 3 · · · are finite non-empty sets, then n =1 A n is finite and non-empty. (c) A ( B C ) = ( A B ) C . (d) A ( B C ) = ( A B ) C . (e) A ( B C ) = ( A B ) ( A C ). Partial Solution. (a) is false: consider A n = { n, n + 1 , n + 2 , . . . } ⊆ N for each n . Then certainly the A n are infinite and nested as desired, but n =1 A n = . (b) is true, but surprisingly delicate to prove. (We were not asked to prove it.) Compare with Theorems 1.4.1 and 3.3.5. (c) is false; (d) and (e) are true, with elementary proofs. Problem 2 (Exercise 1.2.7) . Recall that for any function f : D R and B R , by definition f - 1 ( B ) = { x D | f ( x ) B } . (a) Let f ( x ) = x 2 , A = [0 , 4], and B = [ - 1 , 1]. Find f - 1 ( A ) and f - 1 ( B ). Does f - 1 ( A B ) = f - 1 ( A ) f - 1 ( B )? Does f - 1 ( A B ) = f - 1 ( A ) f - 1 ( B )?

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01sol - ASSIGNMENT 1 SOLUTIONS MAT 472 FALL 2011 Problem...

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