The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

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 $(b, a) \in R$.

Let $R_{1}$ and $R_{2}$ be relations on the set $\{1, 2, \ldots, 50\}$ such that $R_{1} = \{(p, p^{n}) : p \text{ is a prime and } n \geq 0 \text{ is an integer}\}$ and $R_{2} = \{(p, p^{n}) : p \text{ is a prime and } n = 0 \text{ or } 1\}$. Then,the number of elements in $R_{1} - R_{2}$ is:

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

Let $S = \{1, 2, 3, \ldots, 10\}$. Suppose $M$ is the set of all subsets of $S$. Then the relation $R = \{(A, B) : A \cap B \neq \phi; A, B \in M\}$ is :

For the real numbers $x$ and $y$, we define the relation $p$ as $x p y$ if $x-y+\sqrt{2}$ is an irrational number. Then the relation $p$ is