Q3: Is $\{(a, b) : \sqrt a = b\}$ a function? Or: union and intersection of equivalence classes

A relation in R is a function only when every real number $a$ has exactly one image $b$⁠.

The square of a real number is never negative, so $\sqrt{a}$ is a real number only when $a \ge 0$⁠. For example, no real $b$ satisfies $b = \sqrt{-4}$⁠, so $-4$ has no image.

Every element $a$ of A lies in its own class, because $(a, a) \in R$ (R is reflexive). So the classes together cover all of A.

Two different equivalence classes never share an element. So no element is in all three of $A_1, A_2, A_3$⁠, and their intersection is the empty set $\phi$⁠.

Example: on $\{1, 2, 3, 4, 5, 6\}$⁠, "$a - b$ is divisible by 3" gives the classes $\{1, 4\}$⁠, $\{2, 5\}$⁠, $\{3, 6\}$⁠.