An Equivalence Relation is a Binary Relation that is Reflexive, Symmetric and Transitive.
Reflexive: a ~ a
Symmetric: if a ~ b, then b ~ a
Transitive: if a ~ b and b ~ c, then a ~ c
Appendix A
Alternately, we could write, an Equivalence Relation is a Binary Relation that has Reflexivity, Symmetry and Transitivity. Notice the change includes going from “is” to “has”.
Appendix B
Note, if we have
if a ~ b, then b ~ a
then we automatically have
if b ~ a, then a ~ b
Having both of these gives us
a ~ b if and only if b ~ a
An author might write the if and only if statement rather than write the if, then.
Rob Sterling 