   Chapter 1.2, Problem 14E

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Textbook Problem
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# Let f : Z → { − 1 , 1 } be given by f ( x ) = { 1 if   x   is   even − 1 if   x   is   odd a. Prove or disprove that f is onto.b. Prove or disprove that f is one-to-one.c. Prove or disprove that f ( x 1 + x 2 ) = f ( x 1 ) f ( x 2 ) .d. Prove or disprove that f ( x 1 x 2 ) = f ( x 1 ) f ( x 2 ) .

a)

To determine

The proof of the statement “f is onto” is true or not.

Explanation

Given Information:

Let f:Z{1,1} be given by f(x)={1ifxiseven1ifxisodd

Explanation:

Let f:XY

Function f is called onto if and only if f(x)=y.

Let f:Z{1,1} be given by f(x)={

b)

To determine

The proof of the statement “f is one-to-one” is true or not.

c)

To determine

The proof for the statement f(x1+x2)=f(x1)f(x2) is true or not.

d)

To determine

The proof of the statement “f(x1x2)=f(x1)f(x2)” is true or not.

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