Consider the following relations:
$${R_1}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is even over the set of integers
$${R_2}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is odd over the set of integers
$${R_3}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,a.b > 0$$ over the set of non-zero rational numbers
$${R_4}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left| {a - b} \right| \le 2$$ over the set of natural numbers

Which of the following statements is correct?

The number of binary relations on a set with $$n$$ elements is:

Let $${R_1}$$ and $${R_2}$$ be two equivalence relations on a set. Consider the following assertions:

(i)$$\,\,\,\,{R_1} \cup {R_2}$$ is an euivalence relation
(ii)$$\,\,\,\,{R_1} \cap {R_2}$$ is an equivalence relation

Which of the following is correct?

Suppose $$A$$ is a finite set with $$n$$ elements. The number of elements in the Largest equivalence relation of $$A$$ is