Definition of terms for functions

Question

Hi I have this question from my homework and I'm stuck on it. It says:

Let X and Y be sets. Suppose that X is finite and let f : X → Y be a map. Show that f is injective if and only if |f(X)| = |X|.

Firstly what do the modulus signs around f(X) mean and what does id_X mean in terms of functions.

Ok thanks for the quick responses. So in the question where it says "|f(X)| = |X|" does that mean the number of elements in f(X)= to the number of elements in X

$|X|$ is the cardinality of the set $X$ : when $X$ is finite it is simply the number of elements of $X$. — Mauro ALLEGRANZA, Nov 09 '16 at 13:42
$\text {id}_X$ is the identity function on the set $X$, i.e. the function that maps every $x \in X$ into itself. — Mauro ALLEGRANZA, Nov 09 '16 at 13:43

score 2 · Accepted Answer · answered Nov 09 '16 at 13:43

The modulus signs mean "the size of", or "the cardinality of". For instance, $|\{1,2,7,10\}|=4$.

If $X$ is infinite, then $|X|$ still works; there are infinite cardinalities. Beware, though. Intuition might not be your friend any more when dealing with infinite cardinalities.

$\operatorname{Id}_X$ is the unique function $X\to X$ given by $\operatorname{Id}_X(x)=x$. It's called the identity function on $X$.

Definition of terms for functions

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