   Chapter 7.4, Problem 2ES ### 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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# Show that “there are as many squares as there are numbers” by exhibiting a one-to-one correspondence from the positive integers, Z + , to the set S of all squares of positive integers: S = { n ∈ Z + | n = k 2 , for some positive integer k}.

To determine

To show:

“there are as many squares as there are numbers” by exhibiting a one-to-one correspondence from the positive integers, Z+, to the set S of all squares of positive integers.

Explanation

Given information:

S={nZ+|n=k2, for some positive integer k}

Proof:

Let S = Set of all squares of positive integers

We need to define a one-to-one correspondence from Z+ to S.

Let us define the function f as:

f:Z+S,f(x)=x2

f onto:

Let yS.

By the definition of S, there exists a positive integer x such that:

y=x2

By the definition of f :

y=x2=f(x)

We then note that for every yS, there exists a positive integer xZ+ such that y=f(x). By the definition of onto, f is then onto

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