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 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.

Given information:

S={n∈Z+|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 y∈S.

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 y∈S, there exists a positive integer x∈Z+ such that y=f(x). By the definition of onto, f is then onto