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The problem discusses the *power set* of a set \( X \), which is the set of all subsets of \( X \). For example, if \( X = \{a, b\} \), the power set of \( X \), denoted as \( \mathcal{P}(X) \), is:

\[
\mathcal{P}(\{a, b\}) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}.
\]

The problem then asks you to prove the general case:

**For every positive integer \( n \), show that a set with exactly \( n \) elements has a power set with exactly \( 2^n \) elements.**

### Solution:

#### Step 1: Understanding Subsets
For a set \( X \) with \( n \) elements, each element of the set can either be included or not included in a subset. Each element thus has two possibilities (either in the subset or not in the subset).

#### Step 2: Total Number of Subsets
For \( n \) elements, the number of ways you can form subsets is determined by the choices for each element:
- For element 1: either include it or not.
- For element 2: either include it or not.
- And so on, up to element \( n \).

Thus, there are \( 2 \times 2 \times \cdots \times 2 = 2^n \) different subsets.

#### Step 3: Formal Argument
Each subset of a set \( X \) corresponds to a distinct binary sequence of length \( n \), where:
- A "1" in the sequence indicates that the corresponding element of \( X \) is included in the subset.
- A "0" in the sequence indicates that the corresponding element is not included in the subset.

Since there are \( 2^n \) possible binary sequences of length \( n \), the total number of subsets (including the empty set) is \( 2^n \).

#### Conclusion:
Thus, the power set of a set with \( n \) elements contains \( 2^n \) elements.

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Would you like any more details on this proof, or would you like me to explain another part in more depth?

Here are 5 related questions to explore:
1. What is the power set of the set \( \{1, 2, 3\} \)?
2. How does the number of elements in a set affect its power set?
3. Can a power set be empty? Why or why not?
4. How does the power set of an infinite set differ from a finite one?
5. How can you construct the power set of a set with three elements?

**Tip:** The empty set \( \emptyset \) is always an element of every power set!

For every positive integer n, show that a set with exactly n elements has a power set with exactly 2^n elements.

Understand how to prove that for every set with n elements, the power set contains exactly 2^n elements. This problem explores fundamental concepts in set theory and combinatorics, making it ideal for advanced high school or undergraduate students.

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