What's the relationship of this definition and the injective function?

Question

I have the following definition:

Let $f$ be a function. We say that $f$ is injective if $(a, y)\in f$ and $(b,y)\in f$ (i.e., $f(a) = f(b)$) then $a = b$.

I understand the last sentence but I cannot establish the relationship between the definition and injective function.

Answer 1

If $$ a=b \implies f(a)=f(b) $$

Then $f$ is a function.

If $$ f(a)=f(b) \implies a=b $$

then $f$ is injective.

For example $$f=\{ (1,2), (1,3)\}$$ is not a function.

$$f=\{ (1,2), (3,2)\}$$ is a function but it is not injective.

$$f=\{ (1,2), (2,3)\}$$ is a function. It is also injective.

Answer 2

This definition says that no two distinct points are mapped to the same value. You can rewrite this in terms of the contrapositive statement, saying that $f$ is injective iff $a\neq b \implies f(a) \neq f(b)$ for all $a,b$ in the domain.