The relation $R = \{(a, b) : \operatorname{gcd}(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$ is:

Let $A = \{-2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $x R y$ if and only if $y = \max \{x, 1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added to $R$ to make it reflexive and symmetric,respectively. Then $l + m + n$ is equal to

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:

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\alpha R\beta \Leftrightarrow \alpha \perp \beta$,where $\alpha, \beta \in L$. Then $R$ is

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $xv = yu$. Show that $R$ is an equivalence relation.

If $R$ is the smallest equivalence relation on the set $\{1, 2, 3, 4\}$ such that $\{(1, 2), (1, 3)\} \subset R$,then the number of elements in $R$ is