Ordnungen und Hasse-Diagramme
About this app
Ordnung ohne Zahlen: Das vollständige Pfeilbild von Anziehen, Teilmengen oder Pfannkuchenbacken faltet sich in drei Stufen zum Hasse-Diagramm – Schleifen und Abkürzungen verschwinden, vorher stimmt die Klasse ab, welche. Minimale und maximale Elemente leuchten, zwei angetippte Dinge zeigen ihre Kette oder dass sie unvergleichbar sind. Im Modus Reihenfolge hängt die Klasse alles auf eine Leine (topologisches Sortieren), Verstöße blinken.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Partial order and Hasse diagram
A partial order is a relation that is reflexive, antisymmetric and transitive. An example without numbers is getting dressed: socks have to go on before shoes, trousers before the belt. Drawing all arrows gives a dense arrow diagram with a loop at every element and many shortcuts. The Hasse diagram leaves out both, because reflexivity and transitivity supply them anyway. It shows only the direct successors, from bottom to top and without arrowheads. Two elements are comparable if an upward path connects them. If any two are comparable, the order is called a total order.
What the app shows
Three orders are available: getting dressed, the subsets of circle, triangle and square with the subset relation, and making pancakes. The complete arrow diagram folds into the Hasse diagram in three animated stages: first the loops shrink, then the shortcuts light up and disappear, finally the elements glide onto levels. Before that a card asks which arrows will disappear. Minimal and maximal elements light up on request, and two tapped elements show their upward chain or that they are incomparable.
Try it yourself
In sequence mode you drag the clothes or cooking steps onto a clothesline. An element only hangs if everything below it is already there, otherwise the missing predecessors blink. The result is a total order that respects the partial order, a topologically sorted sequence. Arcs below the line then all point to the right, and the app counts how many different sequences you find. Under Practice there are tasks to call out: find the minimal elements, look for an incomparable pair, recognise a least element. The app is built for the projector, the class calls out.
Frequently asked questions
- What is a Hasse diagram?
- A Hasse diagram draws a finite partial order as sparingly as possible. It shows only the direct successors as lines from bottom to top. Loops and shortcuts are left out because they follow from reflexivity and transitivity. An element lies below another if an upward path leads from it to the other.
- What is the difference between a minimal and a least element?
- Nothing lies below a minimal element. A least element lies below all others. When getting dressed, underpants, socks, T-shirt and watch are minimal, but none lies below all, so there is no least element. For the subsets of a set, the empty set is the least element.
- What does topological sorting mean?
- Topological sorting puts the elements of a partial order into a row so that every element comes after all its predecessors. For finite partial orders this always works: take a minimal element of what is left each time. Usually there are many allowed sequences.
- What can I try in the app?
- Fold the arrow diagram into the Hasse diagram and vote beforehand which arrows disappear, light up minimal and maximal elements, compare two elements, hang clothes or cooking steps on a line in an allowed sequence and solve tasks to call out.
Subject: Mathe ohne Zahlen
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- Klassenrunde: Abstimmen, Losen, Zeit
- Königsberger Brücken
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- Relationen-Labor
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