   Chapter 7.2, Problem 13ES ### 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 H :   R → R by the rule H ( x ) = x 2 , for each real number x.(i) Is H one-to-one? Prove or give a counterexample.(ii) 1s H onto? Prove or give a counterexample. b. Define K :   R n o n n e g → R n o n n e g by the rule K ( x ) = x 2 , for each nonnegative real number x. Is K onto? Prove or give a counterexample.

To determine

(a)

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

Explanation

Given information:

Define H:RR by the rule, H(x)=x2 for each real number x.

Calculation:

Let us suppose m,nR and assume that H(m)=H(n) and now prove m=n

So from the definition of H(x), the expression can be written as follows-

m2=n2m=±n

This shows that the values of domain will be different even if outputs are samebecause the square of any negative integer is also positive.

Thus two different values of xR when substituted into the function H(x)=x2 may give the same value.

Since the same output is possible for two different inputs so it is proved that H(x)=x2 where xR is not one-to-one.

For the function to be onto there must be some real value yR in the codomain for which there should be some real integer in the domain of the function

To determine

(b)

To prove K is onto otherwise give counterexample.

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