Intuitive definition of injective, surjective and bijective

Question

I tried to find a more intuitive way of explaining to myself how injective and surjective functions work. Does the following make sense? I'm assuming you have a function defined in the form $f(x)=y$.

1. Injective functions, for every unique $y$-value, have at most one corresponding $x$.

2. Surjective functions, for every unique $y$-value, have at least one corresponding $x$.

3. Bijective functions are both injective and surjective, so for every unique $y$-value, they have exactly one corresponding $x$.

Answer 1

Here is another angle for injectivity:

• "$f:X \to Y$ is a function" is equivalent to "$x_1=x_2 \implies f(x_1)=f(x_2)$"
• "$f:X \to Y$ is injective" is equivalent to "$x_1=x_2 \impliedby f(x_1)=f(x_2)$"

Additionally, when counting the finite number of elements in sets $S$ and $R$, we have by pigeonhole principle:

• If $f:S \to R$ is injective, then $R$ has at least as many elements as $S$; $|S| \leq |R|$.
• If $f:S \to R$ is surjective, then $S$ has at least as many elements as $R$; $|S| \geq |R|$.
• If $f:S \to R$ is bijective, then $S$ and $R$ have exactly the same number of elements; $|S|=|R|$.

Also:

• Function: for each input there exists a unique output.
• Bijective function: for each output there exists a unique input (and vice versa!). Existence from surjectivity, uniqueness from injectivity. This gives a sense of why a bijection is sometimes called a "one-to-one correspondence."

Answer 2

For the function $$ f(x) = 2 $$ what is your $y$? The $2$ or else?

That is one reason why one provides the sets the function maps: $$ f : X \to Y $$ One characterization of a surjective function would be $f(X) = Y$.