The number of relations, defined on the set ${a, b, c, d}$, which are both reflexive and symmetric, is equal to:

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N \times N$ defined by $(a, b) R (c, d)$ if $ad(b + c) = bc(a + d),$ 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 $\mathbb{Z} \times \mathbb{Z}$ defined by $(a, b) R (c, d)$ if and only if $ad - bc$ is divisible by $5$. Then $R$ 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 $A = \{0, 3, 4, 6, 7, 8, 9, 10\}$ and $R$ be the relation defined on $A$ such that $R = \{(x, y) \in A \times A : x - y \text{ is an odd positive integer or } x - y = 2\}$. The minimum number of elements that must be added to the relation $R$ so that it becomes a symmetric relation is equal to $...........$.