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.

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!