Proving Square is an Infinite Set

[amiv4]

Homework Statement

Prove that Square = {x ∈ N|∃y : N. x = y^2} is an infinite set.

Homework Equations

Definition 0.1. A set A is infinite if there is an injection f : A → A that is not also a surjection.

The Attempt at a Solution

I get how to show that there is an injection, but i don't get how to show that it is also not a surjection

 

[Office_Shredder]

What injection are you looking at?

 

[amiv4]

to prove the injection i thought u could do something like this

f(x) = f(y) for arbitrary x and y in N, then x^2 = y^2 and so x = y. and that proves an injection

 

[Mark44]

Homework Statement

Prove that Square = {x ∈ N|∃y : N. x = y^2} is an infinite set.

What is the above supposed to be saying?

Homework Equations

Definition 0.1. A set A is infinite if there is an injection f : A → A that is not also a surjection.

The Attempt at a Solution

I get how to show that there is an injection, but i don't get how to show that it is also not a surjection

 

[amiv4]

would it be that like 0 maps to 0, 1 to 1, 2 to 4, 3 to 9... so that shows injection but then like 5 doesn't map back to anything

 

[Office_Shredder]

Except you're supposed to have a map that goes from the set of squares to the set of squares. But the function that you're looking at will still work

 

[amiv4]

so would it be like 4 mapped to 16 is an injection but if you go the other way 16 doesn't map back to 4 and that is why it isn't a surjection

 

[Mark44]

What set is this? {x ∈ N|∃y : N. x = y^2}

Some characters are rendering as empty boxes.

 

[amiv4]

it is bascially the set of square numbers