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.