One-One (Injective) and Onto (Surjective) Functions

Two independent properties refine what it means for a function to relate its domain and co-domain 'nicely'.

One-one (injective) function. A function $f:A\to B$ is said to be one-one if different elements of $A$ always have different images in $B$. Formally, $f(a)=f(b)\implies a=b$ (equivalently, the contrapositive: $a\ne b\implies f(a)\ne f(b)$). Pictorially, no two arrows from $A$ ever land on the same point of $B$ (Fig. 6.4, Fig. 6.5).

Onto (surjective) function. A function $f:A\to B$ is said to be onto if every element $y$ of $B$ is the image of some $x$ in $A$ — every element of $B$ has at least one pre-image in $A$. The set of all actual images, $f(A)=\{y\in B\mid y=f(x)\text{ for some }x\in A\}$, is called the range of $f$. Saying $f$ is onto is exactly the same as saying $f(A)=B$, i.e. the range fills up the whole co-domain. In general the range is always a subset of the co-domain: range of $f=f(A)\subseteq$ co-domain of $f$.

These two properties are independent — a function can have neither, either one alone, or both:

• One-one but not onto (Fig. 6.4): every arrow lands on a different point, but some element of $B$ (labelled 5 in the figure) receives no arrow at all.
• One-one and onto (Fig. 6.5): every arrow lands on a different point, and every element of $B$ receives exactly one arrow — the two sets are matched up perfectly. …