Relationen-Labor
About this app
Relationen als Pfeile zwischen Figuren und synchron als Gitter M × M: Reflexiv, symmetrisch, antisymmetrisch und transitiv leuchten als Lampen und zeigen ihr Muster, fehlende Pfeile blinken gestrichelt. Hüllen wachsen herein, die transitive Welle für Welle nach einer Vorhersage der Klasse; dazu Alltagsvorlagen, Aufgaben und drei Beweise (Schnitt transitiver Relationen, R = R⁻¹, der Fehlschluss symmetrisch + transitiv ⇒ reflexiv).
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What a relation is
A relation on a set M says which elements are related to each other. Each arrow a → b is an ordered pair (a, b), and the relation is the set of these pairs, a subset of the Cartesian product M × M. Reflexive means every element is related to itself. Symmetric: every arrow has its reverse arrow. Antisymmetric: between two different elements there are never arrows both ways. Transitive: for every path a → b → c there is also the shortcut a → c. Antisymmetric is not the opposite of symmetric; a relation can be both or neither.
What the app shows
The elements are figures instead of numbers: circle, triangle, square, star, diamond, heart and hexagon. On the left they sit on a circle, connected by arrows and loops; on the right the same relation appears as a grid M × M, row for from, column for to. Four lamps show which properties hold. Every property is a visible pattern: reflexive is the full diagonal, symmetric the mirror image in the diagonal, transitive the shortcut for every path. Missing or offending arrows blink dashed.
Try it yourself
Drag arrows from figure to figure, double tap for a loop or switch cells in the grid. Load everyday templates such as is a sibling of, knows, is bigger than or has the same colour as. With one button the reflexive, symmetric and transitive closures grow in, the transitive one wave by wave; beforehand the class predicts whether a few, many or all arrows will be added. The proof mode shows step by step why the intersection of transitive relations is transitive, why R is symmetric exactly when R equals R⁻¹, and where the fallacy symmetric and transitive implies reflexive breaks.
Frequently asked questions
- What is the difference between symmetric and antisymmetric?
- Symmetric means that for every arrow a → b there is also b → a. Antisymmetric means that for different a and b there are never both arrows. This is not the opposite: a relation consisting only of loops is both, a relation with a two-way pair and a one-way arrow is neither.
- What is the transitive closure of a relation?
- The transitive closure is the smallest transitive relation that contains the given one. You add the shortcut a → c for every path a → b → c and repeat until nothing is missing, because new shortcuts form new paths.
- Does symmetric and transitive imply that a relation is reflexive?
- No. The step from a → b via b → a to a → a needs some b that a is related to. An element without a partner gets no loop; the empty relation is symmetric and transitive but not reflexive.
- What can I try in the app?
- Draw arrows and loops, load everyday templates, show the pattern and violations of each property, build closures, estimate the number of new arrows beforehand, solve tasks and follow three proofs about relations step by step.
Subject: Mathe ohne Zahlen
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