Properties of Relations

Intro to Proofs · Axiom Academy

LESSON Properties of Relations Reflexive, symmetric, antisymmetric, and transitive — each one is a pattern you can see in the arrows of a directed graph. For every element , the pair (a,a) must be in R . In the directed graph that means every vertex carries a self-loop — no exceptions. Symmetry means the relation "goes both ways." In the graph, every arrow has its reverse : draw and the partner must appear too, so the edges come in matched pairs. Antisymmetric is not the opposite of symmetric. It forbids a two-way arrow between distinct elements: if and both exist with , the property is broken. A self-loop (a,a) is always fine. Transitivity is "chaining." In the graph, whenever a path exists, the shortcut arrow must exist too — every two-step route comes with a one-step copy. Every relation can be read two ways — as a directed graph or as a 0/1 matrix — and the properties show up in both . Each arrow is exactly a 1 in row i , column j . You've seen each property of a relation as an arrow pattern in a directed graph — and how the same patterns appear in its 0/1 matrix. Scroll up to revisit any step.

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