Find the number of equivalence relations on the set $\{1,2,3\}$ containing $\left(1,2\right)$ and $\left(2,1\right)$.

Show that the relation $R$ defined on the set $A$ of all polygon in a plane as $R=\left\{\left(P_1,P_2\right):P_1 \mathrm{and} P_2 \mathrm{have} \mathrm{same} \mathrm{number} \mathrm{of} \mathrm{sides}\right\}$ is an equivalence relation.

What is the set of elements in $A$ related to the right angled triangle $T$ with sides $3,4$ and $5$?

For the partition $P=\left\{\left\{a\right\}, \left\{b,c\right\}, \left\{d,e\right\}\right\}$, write the corresponding equivalence relation on the set $A=\left\{a,b,c,d,e\right\}$.

Check the relation $\rho$ is an equivalence relation on the set of integers.

$a \rho b$ iff $a^2-b^2$ is a multiple of $7$

Check the relation $\rho$ is an equivalence relation on the set of integers.

$a \rho b$ iff $a=b$

Check the relation $\rho$ is an equivalence relation on the set of integers.

$a \rho b$ iff $a\le \left|b\right|$

Check the relation $\rho$ is an equivalence relation on the set of integers.

$a \rho b$ iff $a-b$ is an even integer

Check the relation $\rho$ is an equivalence relation on the set of integers.

$a \rho b$ iff $a\le b$