Relationship Mapper

Intro to Proofs · Axiom Academy

Every relation in mathematics is really just a set of arrows. Draw the arrows, and the deepest properties show up as patterns you can see. When a mathematician says two things are "related" — equals , less than , divides , is a friend of — they mean something exact and surprisingly simple: a relation is just a set of ordered pairs . Pick the pairs that count as related, and you have built the whole thing. Picture four objects . Each time one is related to another we draw an arrow between them, and we record that as an ordered pair. Watch the arrows go up one at a time — the set R on the right grows with them. That set is the relation. An arrow from a to b is the ordered pair (a,b) — it says " a is related to b ," which need not mean " b is related to a ." The relation is nothing more or less than this set of pairs. Two properties you can see : reflexive & symmetric Mathematicians sort relations by the patterns their arrows make. Toggle a property and watch the arrows rearrange. Reflexive means every object relates to itself — a loop at every node. Symmetric means every arrow has a twin pointing back. Pick either, both, or neither. Notice the rule behind each pattern: reflexive means for every object x ; symmetric means . Transitive: chains demand shortcuts

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