   Chapter 7.1, Problem 48ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 41-49 let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y. Determine which of the properties are true for every fubnction F from X to Y of the properties are ture for every function F from X to Y and which are false for at lest for at lest one functions F from X to Y. Justify your answer. For all subests C and D of Y,. F − 1 ( C − D ) = F − 1 ( C ) − F − 1 ( D ) .

To determine

To check the given property whether it is true for all functions F from X to Y or it is false for at least one function F from X to Y.

Explanation

Given information:

C and D are any two subsets of Y, so CY and DY.

Calculation:

Let F be a function from X to Y.

To prove,

F1(CD)=F1(C)F1(D).

The proof can be divided into two parts.

Part - 1: F1(CD)F1(C)F1(D)

Part - 2: F1(CD)F1(C)F1(D)

Part-1:- The proof of the part- 1 is same as

If xF1(CD), then xF1(C)F1(D).

By the definition of inverse image, xX such that F(x)CD

That means, xX such that F(x)Cand F(x)D.

By the definition of inverse image,

xF1(C)andxF1(D)

By the definition of difference of two sets,

xF1(C)F1(D)

Hence,

F1(CD)F1(C)F1(D)

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