Relations Tutorial - Properties of Relations

by David Muñoz and Will Ptacek

Reflexivity:
When all ordered pairs in a relation have the same element in both positions for all elements.
In math: for relation r on A, r is reflexive if:

(x,x) ∈ r for any x ∈ A

Symmetry:
When a relation has the ordered pair (x,y), it also has the ordered pair (y,x)
In math: for relation r on A, r is symmetric if:

(x,y) ∈ r => (y,x) ∈ r for any x, y ∈ A

Transitivity:
When a relation has ordered pairs that can be used to equal others; if (x,y) and (y,z) are in the relation, then (x,z) must also be in the relation.
In math: for relation r on A, r ⊆ A x A is transitive if:

(x,y) ∈ r and (y,z) ∈ r => (x,z) ∈ r for any x, y, z ∈ A