Suppose that $f:X \rightarrow Y$ is a function.
Then an injection can be defined as:
$\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$
Why isn't it defined instead as follows:
$\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Leftrightarrow x_1=x_2$
I think the above statement also captures the situation described by the definition of an injection in words, which is:
If no element of $Y$ is assigned to more than one elementof $X$, i.e. the function takes a different value for each point of the domain.
I can see that the definition is missing the phrase "if and only if". So is that the only reason we don't write "$\Leftrightarrow$"?