# 74hwsol3 - Solutions to Homework Assignment 3 Math 74, Fall...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions to Homework Assignment 3 Math 74, Fall 2006 October 1, 2006 1. (a) Proposition 1. Let f : X Y . Then f is surjective if and only if f ( f- 1 [ B ] ) = B for every subset B Y . Proof. First, assume that f is surjective. Choose an arbitrary subset B Y . We will show that f ( f- 1 [ B ]) = B . From Problem 7(b) on Homework Assignment 2, we have that f ( f- 1 [ B ]) B . Hence it suffices to show that B f ( f- 1 [ B ]). To that end, choose an arbitrary point y B . Since f is surjective, there exists x X such that y = f ( x ). Clearly x { z | f ( z ) B } = f- 1 [ B ] . Thus y = f ( x ) f ( f- 1 [ B ]). This demonstrates that B f ( f- 1 [ B ]). Conversely, assume that f ( f- 1 [ B ] ) = B for every subset B Y . Then f ( f- 1 [ Y ] ) = Y. Since X = f- 1 [ Y ], we see that Y = f ( X ), so f is surjective. (b) Proposition 2. Let f : X Y . Then f is injective if and only if f- 1 [ f ( A )] = A for every subset A X . 1 Proof. Suppose that f is injective. We will show that f- 1 [ f ( A )] = A for every subset A X . Choose an arbitrary subset A X . By Problem 7(a) on Homework Assignment 2, we have that A f- 1 [ f ( A )]. So if suffices to show that f- 1 [ f ( A )] A . To that end, choose a point x f- 1 [ f ( A )]. Then f ( x ) f ( A ). Thus there exists x A such that f ( x ) = f ( x ). Since f is injective, x = x , and so x A . Thus we have demonstrated that f- 1 [ f ( A )] A . Now suppose that f- 1 [ f ( A )] = A for every subset A X . We will show that f is injective. Choose points x 1 ,x 2 X for which f ( x 1 ) = f ( x 2 ). Then { x 1 } = f- 1 [ f ( { x 1 } )] = f- 1 [ { f ( x 1 ) } ] . Since x 2 f- 1 [ { f ( x 1 ) } ], it follows that x 1 = x 2 . 2. (a) This statement is false. Consider the map f : { 1 } { 1 , 2 } given by 1 7 2. Then f ( f- 1 [ f ( )]) = = f ( ), and f ( f- 1 [ f ( { 1 } )] ) = { 2 } = f ( { 1 } ) ....
View Full Document

## 74hwsol3 - Solutions to Homework Assignment 3 Math 74, Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online