Let's solve each part of the problem.

(i) Number of subsets of set A

If a set $A$ has $n$ elements, the number of subsets of $A$ is given by $2^n$.

Here, $n = 15$, so the number of subsets of set $A$ is: $2^{15} = 32768$

(ii) Cardinality of the power set of A, $P(A)$

The power set $P(A)$ is the set of all subsets of $A$. Therefore, the cardinality (i.e., the number of elements) of the power set is equal to the number of subsets of $A$.

From part (i), we found that the number of subsets is 32768. Hence, the cardinality of the power set $P(A)$ is also: $|P(A)| = 32768$

Summary:

• Number of subsets of A: 32768
• Cardinality of the power set $P(A)$: 32768

Do you want details or have any questions?

Here are some related questions you might find helpful:

1. What is the formula for the number of subsets of a set with $n$ elements?
2. How do you calculate the number of proper subsets of a set?
3. What is a proper subset?
4. How does the power set relate to the concept of subsets?
5. Can a power set ever be an empty set?
6. If set $A$ has 3 elements, what is the number of subsets?
7. What is the difference between a subset and a power set?
8. How does the concept of power sets apply in different branches of mathematics?

Tip: Remember, the number of subsets of a set with $n$ elements is always $2^n$, and this also equals the cardinality of the power set.