https://r2.erweima.ai/mathbot/image/1725915806182yds1ry4c.jpg

Let's analyze the image and solve the problem step by step.

The set \( A \) is given as:

\[
A = \{1, 2, \{3\}, 4; \{5\}\}
\]

You are asked to identify which of the following statements are true or false.

### 1. \( 1 \in A \)
- This statement asks if the element \( 1 \) belongs to \( A \).
- **Answer**: True, because \( 1 \) is an element of \( A \).

### 2. \( 2 \subseteq A \)
- This statement asks if \( 2 \) is a subset of \( A \). For an element to be a subset, all its elements must be in \( A \), but here \( 2 \) is an individual number.
- **Answer**: False, because \( 2 \) is an element, not a set, and can't be considered a subset.

### 3. \( \{4\} \in A \)
- This statement asks if the set \( \{4\} \) is an element of \( A \). 
- **Answer**: False, because \( 4 \) is in \( A \), but \( \{4\} \) (a set containing \( 4 \)) is not.

### 4. \( 2;4 \in A \)
- It seems this statement might be a typographical error, as it should be either a set or an element.
- **Interpretation**: \( 2, 4 \in A \), checking if both 2 and 4 are elements of \( A \).
- **Answer**: True, since both \( 2 \) and \( 4 \) are elements of \( A \).

### 5. \( \{5\} \in A \)
- This statement checks if the set \( \{5\} \) is an element of \( A \).
- **Answer**: True, because \( \{5\} \) is explicitly listed as an element of \( A \).

### 6. \( \{3\} \subseteq A \)
- This statement asks if the set \( \{3\} \) is a subset of \( A \), meaning whether all elements of \( \{3\} \) are in \( A \).
- **Answer**: False, because \( 3 \) is not an element of \( A \); \( \{3\} \) is an element of \( A \).

### 7. \( \{4\} \subseteq A \)
- This statement asks if \( \{4\} \) is a subset of \( A \). 
- **Answer**: True, because the element \( 4 \) is in \( A \), making \( \{4\} \) a subset.

### 8. \( \varnothing \subseteq A \)
- The empty set is always a subset of any set.
- **Answer**: True, because the empty set is a subset of every set.

### Summary of answers:
1. True
2. False
3. False
4. True
5. True
6. False
7. True
8. True

Would you like more details or have any questions? 

Here are five questions to deepen your understanding:
1. What is the difference between an element and a subset?
2. Can a set contain both elements and subsets?
3. Why is \( \varnothing \) always a subset of any set?
4. How does \( \in \) differ from \( \subseteq \)?
5. Can an element of a set also be a set itself?

**Tip**: Remember that sets and their elements can contain other sets, and the relationships between elements and subsets can sometimes be tricky to distinguish.

Given the set A = {1, 2, {3}, 4; {5}}, determine which of the following statements are true:
1 ∈ A, 2 ⊂ A, {4} ∈ A, 2;4 ∈ A, {5} ∈ A, {3} ⊂ A, {4} ⊂ A, ∅ ⊂ A

Analyze set A = {1, 2, {3}, 4; {5}} to determine the truth of various statements involving element membership, subsets, and the empty set. This problem covers core set theory concepts and is ideal for students in Grades 8-10.

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