Function Detective

Intro to Proofs · Axiom Academy

A function is just a rule for sending each input to one output. Watch the arrows — and three big words (injective, surjective, bijective) turn out to be things you can simply see. A function is a machine that turns inputs into outputs Picture two sets of dots: a domain of inputs on the left and a codomain of possible outputs on the right. A function draws exactly one arrow out of every input. The whole subject of how functions behave comes down to a question you can answer by eye: where do those arrows land? Watch one mapping go through three forms. First injective — every arrow lands on its own target, so no two inputs collide. Then surjective — every target on the right gets hit, with nothing left out. Then bijective — both at once: a perfect one-to-one pairing of the two sets. Same dots, same arrows — only where they land changes. That is the entire game of injective, surjective, and bijective. Injective: can you keep the arrows from colliding? A function is injective (one-to-one) when different inputs always give different outputs — visually, no two arrows are allowed to land on the same target. Drag the head of each arrow onto a target on the right. The moment two arrows share a target, the mapping stops being injective. Can you keep all four apart? Different inputs, different outputs — that one-to-one promise is exactly the horizontal-line test you may have seen on graphs. Surjective: can you cover every target?

This is the written version of the interactive lesson above. See the full Intro to Proofs course.