Q5. If n(A) = 3, how many elements does P(P(A)) have?
P(A) has 23 = 8 elements, so P(P(A)) has 28 = 256.
Q6. How many subsets of {1, 2, 3, 4, 5} contain the element 1?
Put 1 in, then choose freely among the other four elements: 24 = 16. That is exactly half of the 32 subsets.
Q7. If A ⊂ B, is P(A) ⊂ P(B)? Explain.
Yes. Every subset S of A is also a subset of B, because each element of S is in A and so in B. So every element of P(A) is in P(B). The converse also holds: if P(A) ⊂ P(B), then A ∈ P(A) ⊂ P(B), so A ⊂ B.
Q8. For A = {1, 2}, which are true: ϕ ∈ P(A), ϕ ⊂ P(A), {1} ⊂ P(A), {{1}} ⊂ P(A)?
All except {1} ⊂ P(A), which fails because 1 is not an element of P(A); its elements are sets. {{1}} ⊂ P(A) holds because {1} ∈ P(A).
Q9. Two finite sets have m and n elements. The number of subsets of the first is 56 more than the number of subsets of the second. Find m and n.
2m − 2n = 56 = 64 − 8 = 26 − 23, so m = 6 and n = 3. (Writing 2n(2m − n − 1) = 8 × 7 shows this is the only solution, since 2m − n − 1 must be odd.)