Injective function

An injective function is a function $f:A\to B$ such that whenever $f(a_1)=f(a_2)$, it follows that $a_1=a_2$.

Injectivity means distinct elements of the domain remain distinct after applying $f$. In terms of cardinality , an injective function from $A$ to $B$ is evidence that $A$ is “no larger than” $B$.

Examples:

• The inclusion map $i:S\to A$ of a subset is injective, since $i(s)=i(s')$ implies $s=s'$.
• The function $f:\mathbb{Z}\to\mathbb{Z}$ given by $f(n)=2n$ is injective, but it is not surjective onto $\mathbb{Z}$.