One of the most important relations in modern mathematics are equivalence relations. By the way, they appear in a large part of mathematics studies. Equivalence Relation In the previous post, we showed some properties such as: reflective; symmetrical; transitive; anti-symmetric. Some of them are necessary to be an equivalence relation. Definition (Equivalence Relations): A relation $\mathcal{R}$ on a set $A$ is said to be an equivalence relation on $A$ if $\mathcal{R}$ is reflexive, symmetric and transitive.