A: Yes. What you need to show is that every element $x$ of $A$ is an element of $B$.
A: What makes a binary operation valid? Are all conditions satisfied for this to be a binary operation $\circ$ from $S \times S$ to $S$?
A: A unary operation $\circ$ from $S$ to $S$ associates with each element $x$ of $S$ an element of $S$. How many ways can we choose the that image element of $x$? Now, if each element makes its choice independently, we multiply the ways of assigning each of the $n$ elements....
A: The way to show that a set is countable is to show that there is a one-to-one mapping of that set to another (known) countable set. So the rational numbers are countable, and so any subset of the rationals is countable. Can you create a one-to-one map to a subset of the rational numbers, for example?