   Chapter 1.2, Problem 11E

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# For the given f : Z → Z , decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f ( x ) = { x 2 if  x  is even 0  if  x  is odd b. f ( x ) = { 0 if  x  is even 2x if  x  is odd c. f ( x ) = { 2 x + 1  if  x  is even x + 1 2  if  x  is odd d. f ( x ) = { x 2 if  x  is even x − 3 2  if  x  is odd e. f ( x ) = { 3 x  if  x  is even 2 x  if  x  is odd f. f ( x ) = { 2 x − 1  if  x  is even 2x if  x  is odd

a)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

Explanation

Given Information:

The f:ZZ where f is given by,

f(x)={x2 if x is even0 if x is odd

Explanation:

Let f:XY

Function f is called one-to-one, if and only if different elements of A always have different images under f.

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

Consider the given function,

f(x)={x2 if x is even0 if x is odd

Let 1 and <

b)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

c)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

d)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

e)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

f)

To determine

Whether the given function f is onto or one-to-one and the proof of the answer.

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