Let $A = \{2, 3, 4, 5, \ldots, 30\}$ and $\simeq$ be an equivalence relation on $A \times A$,defined by $(a, b) \simeq (c, d)$ if and only if $ad = bc$. Then the number of ordered pairs $(c, d)$ which satisfy this equivalence relation with the ordered pair $(4, 3)$ is equal to:

Let $R = \{(a, a)\}$ be a relation on a set $A$. Then $R$ is

If $R$ and $S$ are two non-empty relations on a set $A$,then which of the following statements is false?

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

Let $Z$ be the set of all integers,$A = \{(x, y) \in Z \times Z : (x-2)^{2} + y^{2} \leq 4\}$,$B = \{(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4\}$,and $C = \{(x, y) \in Z \times Z : (x-2)^{2} + (y-2)^{2} \leq 4\}$. If the total number of relations from $A \cap B$ to $A \cap C$ is $2^{p}$,then the value of $p$ is:

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