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 the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \ldots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ are the minimum number of elements required to be added to $R_1$ and $R_2$,respectively,to make the relations symmetric,then $M + N$ equals

Let $A = \{1, 2, 3, 4\}$ and let $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \mid a,b \in R \text{ and } a - b + \sqrt{3} \text{ is an irrational number} \}$. The relation $r$ is

Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then

Let $R_{1}$ and $R_{2}$ be two relations defined as follows:
$R_{1} = \{(a, b) \in \mathbb{R}^{2} : a^{2} + b^{2} \in \mathbb{Q}\}$ and $R_{2} = \{(a, b) \in \mathbb{R}^{2} : a^{2} + b^{2} \notin \mathbb{Q}\}$
where $\mathbb{Q}$ is the set of all rational numbers. Then: