Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $R$ in the set $A$ of human beings in a town at a particular time given by

$R =\{(x, y): x$ and $y$ live in the same locality $\}$

Let $A=\{1,2,3\} .$ Then show that the number of relations containing $(1,2) $ and $(2,3)$ which are reflexive and transitive but not symmetric is four.

The relation "is subset of" on the power set $P(A)$ of a set $A$ is

Determine whether each of the following relations are reflexive, symmetric and transitive :

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as $\mathrm{R}=\{(x, y): 3 x-y=0\}$

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_{1}, P_{2}\right):\right.$ $P _{1}$ and $P _{2}$ have same number of sides $\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3,\,4$ and $5 ?$