The three properties of relations learned first are reflexivity, symmetry, and transitivity.

Reflexivity is an existence property; a possession property. If you contain this entire particular set of pairs, you’re reflexive. If not, then not.

Symmetry and transitivity are implications; closure properties. Now some pairs don’t come for free – they require other pairs be added as well. Suppose you’re having a party. You have two friends who just started dating and are very attached – so if you have one of them over to the party you also have to have the other. That’s symmetry. You have another set of friends, a married couple with a new baby, who can’t get a babysitter. You can have just the husband or just the wife, but if you want to have both of them come, the baby has to come too. That’s transitivity.

In a numerical example: let A = {1,2,3,4,5}. Start building a relation R. Put (1,2) into R. If we want a symmetric R, we must also add (2,1). {(1,2)} is transitive, so for that we don’t need to add anything. However, if we put in (2,1) we must put in (1,1) and (2,2) for R to be transitive; this is called closing R under transitivity.