Reflexive, Symmetric, and Transitive Relations on a Set

A relation from a set A to itself can be though of as a directed graph. We look at three types of such relations: reflexive, symmetric, and transitive.

A relation is reflexive if every element relates to itself, that is has a little look from itself to itself.
A relation is symmetric if whenever x relates to y, then y relates to x. This looks like every path between x and y has a path back.
A relation is transitive if whenever xRy and yRz, then xRz (this shorthands is read "x relates to y" and so on). This looks like every two step path has a corresponding 1 step path.

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