In the set of all $3 \times 3$ real matrices, a relation is defined as follows: $A$ matrix $A$ is related to a matrix $B$ if and only if there exists a non-singular $3 \times 3$ matrix $P$ such that $B = P^{-1} A P$. This relation is

Let $A = \{1, 2, 3\}$. The relation $R$ on set $A$ is defined as $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Determine the nature of the relation $R$.

The relation $R$ defined on a set $A$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$ for all $a, b \in A$. Based on this definition,the relation $R$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$,which is equivalent to saying that if $a \neq b$,then it is not possible for both $(a, b) \in R$ and $(b, a) \in R$ to be true. Therefore,the condition is that for $a \neq b$,we cannot have both $(a, b) \in R$ and $(b, a) \in R$.

Given the relation $R = \{(1, 2), (2, 3)\}$ on the set $A = \{1, 2, 3\}$,the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is

Give an example of a relation which is reflexive and transitive but not symmetric.

Define a relation $R$ on the interval $[0, \frac{\pi}{2})$ by $xRy$ if and only if $\sec^2 x - \tan^2 y = 1$. Then $R$ is :