Question

How to prove x^2 + y^2 is not onto?

Answer 1

\(z=-1\) ?

Answer 2

So I'm supposed to determine whether x^2 + y^2 is a surjection. I know it's not. But can someone explain how to formally prove this?

Answer 3

So I do f(0,-1)?

Answer 4

Your codomain is \(\mathbb{R}\), right?

Answer 5

No it's Z x Z -- > Z

Answer 6

Your codomain is \(\mathbb{Z}\), and thus, your function is not onto if the codomain does contain ALL integers.

Answer 7

note that there is no integers \(x\) and \(y\) such that \(x^2+y^2=-1\).

Answer 8

Yes, but how do I normally prove this ?

Answer 9

Well, a formal disprove is a concrete counterexample to the fact that \(f(x,y)\) is not onto.

Answer 10

Either a rigorous proof that it is onto, or a concrete counterexample that it is not onto.

Answer 11

and what I just showed is a concrete counterexample (to \(f\) being surjective).

Answer 12

Okay. Got it! Thanks!

Answer 13

yw