Please do try the questions by yourself before resorting to these hints!

1. 2² = 4, 2³ = 8, 3² = 9 and 3³ = 27.
2. [X] ↔ X∩B. [X] = [Y] ⇔ X∩B = Y∩B.
3. Bijective.
4. (a, (b,c)) ↔ ((a,b), c).
   Requires |A| = 0, 1 or ∞ (but don't even think of writing infinity like this!).
   Requires |A| = 0, 2 or ∞.
   Currying.
   |C| = 1 or |B||A| = |B|.|A|.
   (f: A+B→C) ↔ (λa.f(0,a), λb.f(1,b)).
5. Suppose that g exists with the given property. a₁ R a₂ ⇒ [a₁] = [a₂] ⇒ p(a₁) = p(a₂) ⇒ g(p(a₁)) = g(p(a₂)) ⇒ q(f(a₁)) = q(f(a₂)) ⇒ [f(a₁)] = [f(a₂)] ⇒ f(a₁) S f(a₂).
   Define g([a]) = q(f(a)). Check that it is well defined and satisfies p ο g = f ο q.
6. Consider Hasse diagrams.
7. (|B|+1)^|A|.
8. Countable union of countable sets.
9. f ↔ {n | f(n) = 1}.
10. Countably infinite, uncountable, finite (=2), uncountable, countable.
11. The interior of each disc contains a point with rational coordinates.
    Circles can be nested arbitrarily.