Let be the set of all triangles in a plane. Show that the relation is an equivalence relation on .

Solution:

Suppose be a relation defined on A. (As given)
If is Reflexive, Symmetric and Transitive, therefore is an equivalence relation.
So now,

Reflexivity:
Suppose be an arbitrary element of
we have,
since, every triangle is similar to itself.
and
Therefore, is reflexive.

Symmetric:
Suppose and , such that

Therefore, is symmetric

Transitivity:
Suppose such that and and


Therefore, is transitive.
As a result, is an equivalence relation.