https://file.aiquickdraw.com/mathbot/file/1730190513320mwyh6g17.jpg

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? 

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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.

Es seien A = {a, b, c, d, e} und B = {M | M ⊂ A}. Man entscheide, welche der folgenden Aussagen wahr, welche falsch sind:

(a) a ∈ B,
(b) {b} ∈ B,
(c) {a} ∈ A,
(d) A ∈ B,
(e) A ⊂ B,
(f) {a} ⊂ A,
(g) ∅ ∈ B,
(h) ∅ ⊂ B,
(i) {∅} ⊂ B.

Analyze statements about elements and subsets for sets A and B, where A = {a, b, c, d, e} and B is the set of all subsets of A. Determine the truth values for statements involving subset and element relationships.

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