Function Mapping Explorer

A function sends each element of one set to exactly one element of another. Tap a left-hand element, then a right-hand one, to draw an arrow. Build a mapping and the explorer tells you whether it is even a function — and if so, whether it is one-one, onto, or both (bijective), with the reason. Add or remove elements on either side, or load a worked example to start.

This mapping

Tap a left element, then a right one, to draw an arrow Tap a left element with an arrow to remove it Each left element may have only one arrow

Domain X Codomain Y

Tap a left element to start an arrow.

Examples:

The rules

A function: every left element has exactly one arrow.
One-one (injective): no two left elements point to the same right element.
Onto (surjective): every right element is pointed to.
Bijective: one-one and onto together.

Why it matters

Only a bijective function can be reversed — it has an inverse. One-one means no information is lost (distinct inputs, distinct outputs); onto means nothing in the target is wasted. A function that is both pairs up the two sets perfectly, one to one. Note two size facts: if |X| < |Y| the function cannot be onto, and if |X| > |Y| it cannot be one-one.