The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is

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

Let $R_1$ and $R_2$ be two relations on a set $A$ , then choose incorrect statement

Let $R$ be a relation on $Z \times Z$ defined by$ (a, b)$$R(c, d)$ if and only if $ad – bc$ is divisible by $5$ . Then $\mathrm{R}$ is

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a,\,b)|a \ge b,\,a,\,b \in R\} $. Then ${R_1}$ is