Äquivalenzklassen und Partitionen
About this app
Zwölf Figuren fliegen nach einem Kriterium wie „gleiche Form“ in weiche Inseln, die Pfeile wachsen innerhalb der Klassen, und die Matrix M × M sortiert sich zur Blockdiagonale. „Gleiche Form oder Farbe“ zeigt, was ohne Transitivität schiefgeht. Inseln lassen sich von Hand ziehen oder einkreisen; der Beweis-Modus führt Schritt für Schritt durch den Satz Äquivalenzrelation ⇔ Partition bis zum Verschmelzen überlappender Klassen.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Equivalence relation, classes and partition
A relation ~ on a set M is an equivalence relation if it is reflexive (a ~ a), symmetric (a ~ b implies b ~ a) and transitive (a ~ b and b ~ c imply a ~ c). The equivalence class [a] contains all elements related to a. The key theorem: the classes are not empty, cover M and are pairwise equal or disjoint, so they form a partition. Conversely, every partition gives the equivalence relation “lies in the same block”, and both ways undo each other.
What the app shows
Twelve figures made of three shapes, two colours and two sizes form the set. Choose a rule such as “the same shape” and the figures fly into soft islands, with arrows connecting all figures of an island. Three lamps show reflexive, symmetric and transitive. Next to it is the matrix M × M: sorted by classes, blocks appear on the diagonal in the colour of the islands. “The same shape or colour” is not transitive: no islands form, and the missing arrow blinks.
Try it and prove it
Before every regrouping the app pauses and asks “What happens next?”, so a class can vote. You can drag figures into other islands or circle new islands with your finger, and the arrows grow along. The proof mode walks through the theorem in eight steps: trying to let two classes overlap produces the arrow chain x, a, c, b, transitivity gives the shortcut, and the islands merge. Under Practice there are tasks to call out.
Frequently asked questions
- What is an equivalence class?
- The equivalence class [a] of an element a is the set of all elements x with x ~ a. For the relation “the same shape” the class of a circle is the set of all circles. By reflexivity a always lies in its own class.
- Why can two equivalence classes not overlap?
- If an element c lies in [a] and in [b], then c ~ a and c ~ b. For every x in [a] we get x ~ a, a ~ c and c ~ b, so x ~ b by transitivity. Hence [a] is contained in [b] and likewise the other way round: the classes are equal.
- How are partitions related to equivalence relations?
- A partition splits a set into non-empty, disjoint blocks that cover everything. The relation “lies in the same block” is reflexive, symmetric and transitive. Conversely, the classes of every equivalence relation form a partition. The two correspond one to one.
- What can I try in the app?
- Choose rules and vote beforehand on how many islands form, sort the matrix by classes, explore a non-transitive relation, drag or circle islands by hand, follow the proof step by step and solve practice tasks for the whole class.
Subject: Mathe ohne Zahlen
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