If $R$ and $R^1$ are equivalence relations on a set $A$, then which of the following is also an equivalence relation?

On $R$, the relation $\rho$ is defined by '$x \rho y$ holds if and only if $x-y$ is zero or irrational'. Then,

Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is

The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ 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 a relation defined on the set $Z$ of all integers such that $x R y$ if and only if $x+2y$ is divisible by $3$. Then: