Let $R = \{(1,2), (2,3), (3,3)\}$ be a relation defined on the set $A = \{1, 2, 3, 4\}$. Then the minimum number of elements needed to be added to $R$ so that $R$ becomes an equivalence relation is:

Let a set $A = A_{1} \cup A_{2} \cup \ldots \cup A_{k}$,where $A_{i} \cap A_{j} = \phi$ for $i \neq j$ and $1 \leq i, j \leq k$. Define the relation $R$ from $A$ to $A$ by $R = \{(x, y) : y \in A_{i} \text{ if and only if } x \in A_{i}, 1 \leq i \leq k\}$. Then,$R$ is

Let $R$ be an equivalence relation on a finite set $A$ having $n$ elements. Then the number of ordered pairs in $R$ is

Let $R$ be a relation on the set $N$ defined by $\{(x, y) | x, y \in N, 2x + y = 41\}$. Then $R$ is

Let $R$ be a reflexive relation on a finite set $A$ containing $n$ elements,and let $R$ contain $m$ ordered pairs. Then,

$A$ relation $R$ defined on the set $A = \{a, b, c\}$ is given by $R = \{(a, a), (b, b), (c, c), (a, c)\}$. This relation is . . . . . . .