Zuordnungsmaschine: injektiv, surjektiv, bijektiv
About this app
Funktionen als Pfeilbild ohne Zahlen: Pfeile zwischen Figuren ziehen, Ringe zählen die Treffer, drei Lampen zeigen injektiv, surjektiv, bijektiv. „Umdrehen“ kehrt alle Pfeile um – nur bei einer Bijektion entsteht wieder eine Funktion. Mit Vorhersage-Karte, Rundreise g∘f = id, Verkettung, Gäste und Stühle und Beweis-Modus in beide Richtungen.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Function, injective, surjective, bijective
A function f from A to B assigns exactly one element of B to every element of A. In the arrow diagram exactly one arrow leaves every element of A. Injective means that at most one arrow arrives in every element of B, two different elements never land on the same target. Surjective means that at least one arrow arrives in every element of B. Bijective means both, so exactly one arrow per target. Exactly then every b has exactly one preimage, and there is an inverse map g with g∘f = id on A and f∘g = id on B.
What the app shows
Two banks with shapes instead of numbers: A on the left, B on the right, arrows in between. Every target gets one ring per arriving arrow, targets hit twice and empty targets light up. Three lamps show whether the function is injective, surjective or bijective. The Reverse button turns all arrows around in an animation: only for a bijection is the result a function again. There is also a round trip for g∘f = id, the composition A to B to C and the scenario guests and chairs.
Try it yourself
Drag arrows from A to B, change the size of the sets and load templates such as injective only or surjective only. Before reversing, the app asks: what happens next? This lets a class vote. In proof mode the theorem bijective if and only if exactly one preimage if and only if invertible runs in ten named steps in both directions. Under Practice there are tasks such as make f injective but not surjective, including impossible ones where the right answer is Impossible.
Frequently asked questions
- What is the difference between injective and surjective?
- Both are read off at the target set. Injective means every target is hit at most once: f(a) = f(a′) implies a = a′. Surjective means every target is hit at least once. A function can be both, only one of them or neither.
- Why is only a bijective function invertible?
- If you reverse all arrows, exactly one arrow must lead back from every target for the result to be a function. If a target was hit twice, there would be two ways back; if a target stayed empty, there would be none. Exactly one arrow per target means bijective.
- What does g∘f = id mean?
- Applying f and then g brings every element of A back to itself. From g∘f = id on A it follows that f is injective, from f∘g = id on B that f is surjective. If both hold, g is the inverse map of f.
- What can I try in the app?
- Drag arrows and watch the lamps, reverse the arrows after voting, start round trips, compose two functions, form pairs with guests and chairs, follow the proof step by step and solve tasks, including impossible ones.
Subject: Mathe ohne Zahlen
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