   Chapter 7.2, Problem 40ES ### 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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# Suppose F :   X → Y is one—to—one.a. Prove that for every subset A ⊆ X ,   F − 1 ( F ( A ) ) = A .b. Prove that for all subsets A 1 and A 2 in X, F ( A 1 ∩ A 2 ) = F ( A 1 ) ∩ F ( A 2 ) .

To determine

(a)

To prove:

F1(F(A))=A for any subset AX.

Explanation

Given information:

The function F:XY is one-to-one function and AX.

Concept used:

f(x)=yx=f1(y).

Proof:

The objective is to prove that for all subsets AX,F1(F(A))=A if F:XY is one to one.

Need to show that F1(F(A))Aand AF1(F(A)).

Suppose that xF1(F(A)), then F(x)F(A).

If x is not in A, then there is some element yA such that xy and F(x)=F(y) but this contradicts the fact that F is one to one

To determine

(b)

To prove:

F(A1A2)=F(A1)F(A2), for all subsets A1 and A2 in X.

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