$$  $$If set A has 15 elements, then

(i) Find the number of subsets of A.

(ii) Find the cardinality of power set of A, P(A).

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.

Explore how to find the number of subsets and the cardinality of the power set of set A with 15 elements. Understand the combinatorial concept of subsets and power sets in set theory. Suitable for advanced high school students.

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