How many reflexive relations are there on a set with $3$ elements?

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

Let $A = \{-2, -1, 0, 1, 2, 3, 4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x + y \le 2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to:

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

Let $f: X \rightarrow Y$ be a function. Define a relation $R$ in $X$ given by $R = \{(a, b) : f(a) = f(b)\}$. Examine if $R$ is an equivalence relation.

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$,show that $R_{1} \cap R_{2}$ is also an equivalence relation.