Let’s analyze the problem step by step.

(a) Is it possible for $B$ to contain exactly one set of cardinality 7?

Step 1: Understanding the set $B$

• $A$ is a set, and $B$ is the power set of $A$. This means that $B$ contains all subsets of $A$.
• The number of subsets of $A$ is $2^{|A|}$, where $|A|$ is the cardinality of $A$ (i.e., the number of elements in $A$).
• Among these subsets, we are particularly interested in those subsets of cardinality 7.

Step 2: Analyzing the problem

For $B$ to contain exactly one subset of cardinality 7, the original set $A$ must have at least 7 elements. The number of subsets of a set $A$ of cardinality $n$, that have cardinality 7 is given by the binomial coefficient $\binom{n}{7}$.

So we require:

$\binom{n}{7} = 1$

This condition implies that $n$ must be exactly 7 because:

$\binom{n}{7} = 1 \quad \text{if and only if} \quad n = 7$

Conclusion for (a)

• Yes, it is possible for $B$ to contain exactly one set of cardinality 7 if and only if $A$ has exactly 7 elements. In this case, the only subset of cardinality 7 is $A$ itself.

---

(b) Is it possible for $B$ to contain exactly two sets of cardinality 7?

Step 1: Reconsidering the number of subsets of cardinality 7

For $B$ to contain exactly two subsets of cardinality 7, we need:

$\binom{n}{7} = 2$

However, the binomial coefficient $\binom{n}{7}$ only takes integer values and increases as $n$ increases. The smallest values for $\binom{n}{7}$ as $n$ increases are as follows:

• $\binom{7}{7} = 1$
• $\binom{8}{7} = 8$
• $\binom{9}{7} = 36$

There is no $n$ such that $\binom{n}{7} = 2$. Therefore, it is impossible for $B$ to contain exactly two subsets of cardinality 7.

Conclusion for (b)

• No, it is not possible for $B$ to contain exactly two sets of cardinality 7.

---

Summary:

• (a) Yes, $B$ can contain exactly one set of cardinality 7 if and only if $A$ has exactly 7 elements.
• (b) No, it is not possible for $B$ to contain exactly two sets of cardinality 7.

---

Would you like more details on any part of the reasoning?

Here are some related questions for deeper understanding:

1. What is the power set of a set, and how is it calculated?
2. How does the binomial coefficient $\binom{n}{k}$ relate to subsets of a given cardinality?
3. Can we calculate the exact number of subsets of a given size for any set?
4. How would the situation change if we asked about subsets of a different cardinality, say 5?
5. What is the total number of subsets in a set of cardinality $n$?

Tip: Understanding how binomial coefficients work is essential when analyzing problems involving subsets. These coefficients give us the number of ways to choose subsets of a specific size from a set.