Answered: Let N be the set of natural numbers and…

Let N be the set of natural numbers and the relation R be defined onN such that R = {(x, y) : y = 2x, x, y ∈ N}.What is the domain, codomain and range of R? Is this relation a function?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

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Question

Let N be the set of natural numbers and the relation R be defined on
N such that R = {(x, y) : y = 2x, x, y ∈ N}.
What is the domain, codomain and range of R? Is this relation a function?

Expert Solution

Step 1

Let N be the set of natural numbers and the relation R be defined on
N such that R = {(x, y) : y = 2x, x, y ∈ N}.

The set of all inputs for a function is called domain and the set of allowable outputs is called codomain of a relation.

Therefore, domain of given relation is set of natural numbers and codomain of given relation is set of natural numbers.

Also, range of a relation is set of all values that f takes.

Now, if x is any natural number, then y=2x is an even natural number.

Therefore, Range of R is set of all even natural numbers.

Also, image of every element of domain set under R is defined and is unique, therefore, given relation is a function.

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