What is the power set of P(pie) of the empty set

The power set \( P(A) \) of a set \( A \) is the set of all subsets of \( A \), including the empty set and \( A \) itself. 

Now, let's go through the process for the empty set:

1. Let the empty set be denoted as \( \emptyset \), which is a set containing no elements.
2. The power set of \( \emptyset \) is the set of all subsets of \( \emptyset \). 
3. The only subset of the empty set is the empty set itself.

Therefore, the power set of the empty set is:

\[
P(\emptyset) = \{ \emptyset \}
\]

So, the power set of the empty set contains one element, which is the empty set.

### Do you want further details or have any questions?

Here are 5 questions to expand on this concept:
1. What is the power set of a set with one element, such as \( A = \{a\} \)?
2. How many elements does the power set of a set with \( n \) elements contain?
3. What is the relationship between the cardinality of a set and its power set?
4. Can a power set itself be empty? Why or why not?
5. How would you compute the power set of \( A = \{1, 2\} \)?

**Tip:** The power set of a set with \( n \) elements always contains \( 2^n \) subsets.

What is the power set of P(pie) of the empty set?

Explore the concept of power sets in set theory by learning how to find the power set of the empty set. This solution explains the mathematical process, involving the key formula for power sets and the relationship between a set and its subsets. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation