Show that the relation $R$ defined in the set A of all triangles as $R =\left\{\left( T _{1},\, T _{2}\right):\, T _{1}\right.$ is similar to $\left. T _{2}\right\}$, is equivalence relation. Consider three right angle triangles $T _{1}$ with sides $3,\,4,\,5, \,T _{2}$ with sides $5,\,12\,,13 $ and $T _{3}$ with sides $6,\,8,\,10 .$ Which triangles among $T _{1},\, T _{2}$ and $T _{3}$ are related?

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,1),\,(2,2),$ $(3,3)$, $(1,2)$, $(2,3)\}$ is reflexive but neither symmetric nor transitive.

Let $n(A) = n$. Then the number of all relations on $A$ is

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$, show that $R_{1} \cap R_{2}$ is also an equivalence relation.

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R =\{(a, b):|a-b|$ is even $\},$ is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to  each other and all the elements of $ \{2,4\}$ are