The void relation on a set $A$ is

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 $P(S)$ denote the power set of $S = \{1, 2, 3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ and $A R_2 B$ if $A \cup B^c = B \cup A^c, \forall A, B \in P(S)$. Then:

In a plane,two points $P$ and $Q$ are related if $OP = OQ$,where $O$ is a fixed point. The relation is:

Give an example of a relation which is transitive but neither reflexive nor symmetric.

Let $R$ be a relation defined on the set $\{1,2,3,4\} \times \{1,2,3,4\}$ by $R = \{((a,b), (c,d)) : 2a + 3b = 3c + 4d\}$. Then the number of elements in $R$ is: