Prove that the function f(x,y) = x - y is not a bijection.

Question

How would we prove that a function f(x, y) = x - y is not a bijection? The function's domain is all positive integers.

I know that a function that is bijective has a one-to-one correspondence where each element of one set is paired with exactly one element of the other set, but I have no idea how to prove this.

Answer 1

To be one-to-one you need that if $f(x) = f(y)$ then $x=y$.

Consider $f(3,2)$ and $f(5,4)$. You have that $f(3,2) = 3-2 = 1 = 5-4 = f(5,4)$, but $(3,2) \neq (5,4)$ so the function is not one-to-one and therefore not a bijection

Answer 2

for $y=x$ you have that $f(x,x)=0$$

So it's not a bijection since you have that

$$f(1,1)=f(2,2)=.....=f(n,n)=0$$

Answer 3

In order for a function to be a bijection the function must be one-to-one and onto.

Note that $$f(4,3)=f(3,2)=1$$

Thus $f(x,y)= x-y$ is not one-to-one, hence it is not a bijection.

Answer 4

$f(1,1) = 0$

$f(2,2)=0$

If the output is 0, what were x and y. You can't tell; there are at least two possibilities. Therefore, it's not a bijection.