   Chapter 7.2, Problem 10ES ### 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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# a. Define f :   Z → Z by the rule f ( n ) = 2 n , for every integer n.(i) Is f one-to-one? Prove or give a counterexample.(ii) Is f onto? Prove or give a counterexample. b. Let 2Z denote the set of all even integers. That is, 2 Z = { n ∈ Z | n = 2 k , for some integer k } . Define h :   Z → 2 Z by the rule h ( n ) = 2 n , for each integer n. Is h onto? Prove or give a counterexample.

To determine

(a)

To prove f is one-to-one and onto otherwise give counterexample.

Explanation

Given information:

Define f:ZZ by the rule, f(n)=2n for all integers n.

Calculation:

Let us suppose x,yZ and assume that f(x)=f(y) and now prove x=y

So from the definition of f(n), the expression can be written as follows-

2x=2yx=y{dividing by 2 on both sides}

Thus,

To determine

(b)

To define h:Z2Z by the rule, h(n)=2n for all integers n and also define whether the function is onto or not.

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