 # What is an equivalence relation?

Question:

What is an equivalence relation?

Show that the relation of 'similarity' on the set $S$ of all triangles in a plane is an equivalence relation.

Solution:

An equivalence relation is one which possesses the properties of reflexivity, symmetry and transitivity.

(i) Reflexivity: A relation $\mathrm{R}$ on $\mathrm{A}$ is said to be reflexive if $(\mathrm{a}, \mathrm{a}) \in \mathrm{R}$ for all a $€ \mathrm{~A}$.

(ii) Symmetry: A relation $R$ on $A$ is said to be symmetrical if $(a, b) \in R$ è $(b, a) \in R$ for all $(a, b) \in A$.

(iii) Transitivity: A relation $R$ on $A$ is said to be transitive if $(a, b) \in R$ and $(b, c) \in R$ è $(a$, c) $€ R$ for all $(a, b, c) \in A$.

Let S be a set of all triangles in a plane.

(i) Since every triangle is similar to itself, it is reflexive.

(ii) If one triangle is similar to another triangle, it implies that the other triangle is also similar to the first triangle. Hence, it is symmetric.

(iii) If one triangle is similar to a triangle and another triangle is also similar to that triangle, all the three triangles are similar. Hence, it is transitive.