Let's address each part of the question.

Part (a)

For an empty set $B = \{\}$:

• The number of elements in an empty set is $0$.
• The power set of a set with $n$ elements has $2^n$ elements.
• Therefore, the power set of the empty set will have: $2^0 = 1$ So, the power set of an empty set contains exactly 1 element (which is the empty set itself).

Part (b)

For a set with $k + 1$ elements:

• The power set of a set with $n$ elements has $2^n$ elements.
• Here, $n = k + 1$, so the power set will have: $2^{k+1}$ elements.

Summary

1. For an empty set, the power set has 1 element.
2. For a set with $k + 1$ elements, the power set has $2^{k+1}$ elements.

Would you like more details on how power sets are constructed, or do you have questions?

Here are 5 follow-up questions to expand on this topic:

1. What is the power set of a set with 2 elements?
2. How would you find the number of subsets of a set with 4 elements?
3. Can a power set ever have fewer elements than the original set?
4. What are the elements of the power set for a set $S = \{1, 2, 3\}$?
5. How does the concept of power sets apply in probability theory?

Tip: For any set with $n$ elements, the power set always contains $2^n$ subsets, including the empty subset and the set itself.