Let a relation $R$ be defined by $R = \{(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)\}$ then ${R^{ – 1}}oR$ is

Let $\mathrm{T}$ be the set of all triangles in a plane with $\mathrm{R}$ a relation in $\mathrm{T}$ given by $\mathrm{R} =\left\{\left( \mathrm{T} _{1}, \mathrm{T} _{2}\right): \mathrm{T} _{1}\right.$ is congruent to $\left. \mathrm{T} _{2}\right\}$ . Show that $\mathrm{R}$ is an equivalence relation.

For real numbers $x$ and $y$, we write $ xRy \in $ $x – y + \sqrt 2 $ is an irrational number. Then the relation $R$ is

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

Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$

In the set $A = \{1, 2, 3, 4, 5\}$, a relation $R$ is defined by $R = \{(x, y)| x, y$ $ \in  A$ and $x < y\}$. Then $R$ is