# HW 9 - MATH 290 HOMEWORK 9 SOLUTIONS Let A and B be sets...

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MATH 290 – HOMEWORK 9 – SOLUTIONS Let A and B be sets, and suppose that f : A B . 1. Prove that for every X A , X f - 1 ( f ( X )). Solution: Let X A and suppose that x X . If we let y = f ( x ), then by deﬁnition, y f ( X ) so that f ( x ) f ( X ). This implies that there is an element of X (namely x ) such that f ( x ) f ( X ). Hence x f - 1 ( f ( X )). 2. Prove that f is injective (that is, one-to-one) if and only if for every X A , X = f - 1 ( f ( X )). Solution: (= ). Suppose that f is injective, and let X A . Since we already know that X f - 1 ( f ( X )), all we need to prove is that f - 1 ( f ( X )) X . Let x f - 1 ( f ( X )). Since x f - 1 ( f ( X )), f ( x ) f ( X ) which means that there is a y X such that f ( x ) = f ( y ) f ( X ). But since f is injective, f ( x ) = f ( y ) implies that x = y . Therefore, x = y X . ( =). We will prove this implication by contrapositive. Suppose that
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