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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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Section
BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Let X and Y be sets, let A and B be any subsets of X, and let F be a function from X to Y. Fill in the blanks in the following proof that F ( A ) F ( B ) F ( A B ) .

To determine

To Fill:

The following proof that F(A)F(B)F(AB).

Explanation

Given information:

Let X and Y be sets, let A and B be any subsets of X and let F be a function from X to Y, and the incomplete proof of the statement F(A)F(B)F(AB).

Calculation:

Given:

X and Y are sets

A and B are subsets of X F is a function from X to Y

To proof: F(A)F(B)F(AB)

PROOF:

Let yF(A)F(B). By the definition of the union, we then know that

  1. yF(A) or yF(B)

First case: yF(A)

By the definition of the image, there exists (b) some xA such that y=F(x).

Since xA and since AAB :

x(c)AB

Thus we then obtain y=F(x) for some xAB and thus y lies in the image of AB

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