Let A=  {8,{8,5},{5},Ø}. Which of the following statements are true? There may be more than one correct answer.

A. Ø ⊆ A
B. 5 ⊆ A
C. 5 ∈ A
D. {5} ∈ A
E. Ø ∈ A
F. 8 ∈ A
G. {5} ⊆ A
H. {8, 5} ∈ A
I. {8, 5} ⊆ A
J. {8} ⊆ A
K. 8 ⊆ A
L. {8} ∈ A

We are given the set \( A = \{8, \{8, 5\}, \{5\}, \emptyset\} \), and we need to evaluate which of the following statements about the set are true. Let’s analyze each option:

### A. \( \emptyset \subseteq A \)
This is **true** because the empty set \( \emptyset \) is a subset of every set.

### B. \( 5 \subseteq A \)
This is **false** because 5 is not a set, so it cannot be a subset of \( A \).

### C. \( 5 \in A \)
This is **false** because the number 5 is not an element of \( A \). However, \( \{5\} \) is an element of \( A \).

### D. \( \{5\} \in A \)
This is **true** because the set \( \{5\} \) is explicitly an element of \( A \).

### E. \( \emptyset \in A \)
This is **true** because the empty set \( \emptyset \) is explicitly an element of \( A \).

### F. \( 8 \in A \)
This is **true** because the number 8 is explicitly an element of \( A \).

### G. \( \{5\} \subseteq A \)
This is **false** because \( \{5\} \) is an element of \( A \), but for it to be a subset, each element inside \( \{5\} \) (i.e., 5) would have to be an element of \( A \), which is not the case.

### H. \( \{8, 5\} \in A \)
This is **true** because \( \{8, 5\} \) is explicitly an element of \( A \).

### I. \( \{8, 5\} \subseteq A \)
This is **false** because for \( \{8, 5\} \) to be a subset of \( A \), both 8 and 5 would need to be elements of \( A \). While 8 is an element, 5 is not, so this is false.

### J. \( \{8\} \subseteq A \)
This is **false** because 8 is an element of \( A \), but \( \{8\} \) being a subset means that 8 would have to be an element of \( A \) in such a way that the set \( \{8\} \) itself is included.

### K. \( 8 \subseteq A \)
This is **false** because 8 is not a set, so it cannot be a subset of \( A \).

### L. \( \{8\} \in A \)
This is **false** because \( \{8\} \) is not an element of \( A \). Only 8, not \( \{8\} \), is an element of \( A \).

### Final Evaluation of the Statements:
- A. True
- B. False
- C. False
- D. True
- E. True
- F. True
- G. False
- H. True
- I. False
- J. False
- K. False
- L. False

Would you like a more detailed explanation of any part of this solution?

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Here are 5 related questions to extend your understanding:
1. What is the difference between being an element and being a subset of a set?
2. Can a set be a subset of itself? Why or why not?
3. How do we determine if an element is a subset of a set?
4. What would the power set of \( A \) look like?
5. How do we interpret sets that contain other sets?

**Tip:** Always remember that elements within a set must match exactly when determining whether a set is a subset or element.

Let A= {8, {8,5}, {5}, Ø}. Which of the following statements are true? There may be more than one correct answer. A. Ø ⊆ A B. 5 ⊆ A C. 5 ∈ A D. {5} ∈ A E. Ø ∈ A F. 8 ∈ A G. {5} ⊆ A H. {8, 5} ∈ A I. {8, 5} ⊆ A J. {8} ⊆ A K. 8 ⊆ A L. {8} ∈ A

Analyze the set A = {8, {8,5}, {5}, Ø} and determine which statements about elements and subsets are true. This problem involves concepts from set theory, including elements, subsets, and the empty set. Suitable for high school students in Grades 9-12.

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