# Create an injection from {a^2+b^2

## Homework Statement

Is it possible to create an injection from {a^2+b^2 l a,b$\in$Q} to {{a+b l a,b$\in$Q}

## Homework Equations

I am not sure about this. The problem is that many combinations of a,b can yield the same number, so how do I tackle this obstacle?

## The Attempt at a Solution

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UltrafastPED
Gold Member
Consider the definition of injection.

the definition says:

x1≠x2 => f(x1)≠f(x2)

But the problem is as said that many combinations give rise to the same numbers in the set.

UltrafastPED
Gold Member
Then perhaps it is not possible!

it certainly should be, since the two sets have same cardinality. I just need help with coming up with a well defined function that does it. Problem is you can't really say something like f(a^2+b^2)=lal+lbl because of the earlier stated.

pbuk
Gold Member
it certainly should be, since the two sets have same cardinality.
Is this necessarily true?

I just need help with coming up with a well defined function that does it.

jbunniii
Homework Helper
Gold Member
It might simplify the problem somewhat to observe that ##\{a+b \, | \, a,b\in \mathbb{Q}\}## is simply ##\mathbb{Q}##.

pbuk