The exercise provides two sets:

• $A = \{a, b, c, d, e\}$
• $B = \{M \mid M \subset A\}$

Here, $B$ represents the set of all subsets of $A$. We are asked to determine whether each statement is true or false.

Let's evaluate each option:

Analysis of Each Statement

1. (a) $a \in B$

   • $B$ contains subsets of $A$. Since $a$ is an element of $A$, it is not a subset itself, but rather an individual element.
   • Result: False

2. (b) $\{b\} \in B$

   • The set $\{b\}$ is a subset of $A$ because $b$ is an element of $A$. Therefore, $\{b\} \in B$.
   • Result: True

3. (c) $\{a\} \in A$

   • $\{a\}$ is not an element of $A$; rather, $a$ itself is an element of $A$. Sets in $A$ do not contain subsets, just elements.
   • Result: False

4. (d) $A \in B$

   • Since $A$ is a subset of itself, $A \in B$.
   • Result: True

5. (e) $A \subset B$

   • For $A \subset B$ to hold, each element of $A$ would need to be in $B$. However, elements of $A$ are not subsets of $A$; they are individual elements. Thus, $A \subset B$ does not hold.
   • Result: False

6. (f) $\{a\} \subset A$

   • $\{a\}$ is indeed a subset of $A$ because $a \in A$.
   • Result: True

7. (g) $\emptyset \in B$

   • $\emptyset$ is a subset of any set, including $A$, so $\emptyset \in B$.
   • Result: True

8. (h) $\emptyset \subset B$

   • The empty set is a subset of every set by definition, so $\emptyset \subset B$.
   • Result: True

9. (i) $\{\emptyset\} \subset B$

   • $\{\emptyset\}$ is a set containing the empty set as an element. Since $\emptyset \in B$, $\{\emptyset\} \subset B$ holds.
   • Result: True

Summary of Answers

• (a) False
• (b) True
• (c) False
• (d) True
• (e) False
• (f) True
• (g) True
• (h) True
• (i) True

Would you like further details on any specific part of this solution?

Here are five related questions to explore further:

1. What is the difference between an element and a subset?
2. Why is the empty set considered a subset of every set?
3. How do we determine if a set is a subset or an element of another set?
4. What properties distinguish power sets from regular sets?
5. How does the concept of subsets change with infinite sets?

Tip: Always differentiate between elements and subsets; an element is a part of a set, while a subset is a set containing elements (or none) from another set.