https://file.aiquickdraw.com/mathbot/file/1730031948051dqfectc1.jpg

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?

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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.

Find the number of elements in the power sets of the following:
(a) An empty set, set B = {}.
(b) A set with 'k + 1' elements.

Explore how to determine the number of elements in power sets for an empty set and a set with 'k + 1' elements. Learn the key formula 2^n to find the cardinality of power sets, along with detailed explanations for each scenario.

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