https://r2.erweima.ai/mathbot/image/17259366648905fxp7lw5.jpg

Let's analyze the question. We are given a set \( A \) defined as:

\[
A = \{ \emptyset, \{1, 2\}, \{1\}, \{\emptyset\}, 1, \{2\} \}
\]

The task is to determine whether the listed statements are true or false and explain why. Let's go through each statement:

### a) \( 1 \in A \)

This statement asks if the element \( 1 \) belongs to the set \( A \).  
- In set \( A \), we see that \( 1 \) is indeed an element.  
- Therefore, **this statement is true**.

### b) \( 2 \in A \)

This statement asks if the element \( 2 \) belongs to set \( A \).  
- In set \( A \), the element \( 2 \) does not appear on its own (we only have \( \{1, 2\} \) and \( \{2\} \), but not \( 2 \) alone).  
- Therefore, **this statement is false**.

### c) \( 2 \subseteq A \)

This asks if the element \( 2 \) is a subset of \( A \).  
- A number cannot be a subset, only sets can be subsets. The element \( 2 \) is not a set, so this is **false**.

### d) \( \{2\} \in A \)

This asks if the set \( \{2\} \) is an element of \( A \).  
- Looking at \( A \), we do see that \( \{2\} \) is one of its elements.  
- Therefore, **this statement is true**.

### e) \( \{2\} \subseteq A \)

This asks if the set \( \{2\} \) is a subset of \( A \).  
- To check if \( \{2\} \subseteq A \), we need to verify whether every element of \( \{2\} \) (which is just \( 2 \)) belongs to \( A \). Since \( 2 \notin A \), \( \{2\} \) is not a subset.  
- Therefore, **this statement is false**.

### f) \( \emptyset \subseteq A \) y \( \emptyset \in A \)

This is a compound statement.  
1. \( \emptyset \subseteq A \): The empty set is always a subset of any set, so this part is true.  
2. \( \emptyset \in A \): The empty set is explicitly listed as one of the elements of \( A \). Therefore, this part is also true.  
- Therefore, **this entire statement is true**.

### g) \( 1 \in A \) y \( \{1\} \in A \)

This is another compound statement:  
1. \( 1 \in A \): As previously discussed in part (a), this is true.  
2. \( \{1\} \in A \): The set \( \{1\} \) is also listed as one of the elements of \( A \), so this is true.  
- Therefore, **this entire statement is true**.

### h) \( \eta(A) = 6 \)

This statement refers to the cardinality of set \( A \), which is the number of elements in the set.  
- Set \( A \) contains 6 elements: \( \emptyset, \{1, 2\}, \{1\}, \{\emptyset\}, 1, \{2\} \).  
- Therefore, **this statement is true**.

### i) \( \eta(P(A)) = 64 \)

This statement refers to the cardinality of the power set \( P(A) \), which is the set of all subsets of \( A \).  
- The formula for the cardinality of the power set is \( 2^n \), where \( n \) is the number of elements in the original set.  
- Since \( A \) has 6 elements, \( P(A) \) will have \( 2^6 = 64 \) elements.  
- Therefore, **this statement is true**.

### j) \( \{1, 2\} \subseteq A \)

This asks if the set \( \{1, 2\} \) is a subset of \( A \).  
- For \( \{1, 2\} \subseteq A \), both elements \( 1 \) and \( 2 \) must belong to \( A \). However, while \( 1 \in A \), \( 2 \notin A \).  
- Therefore, **this statement is false**.

### Final Summary:

- **a)** True  
- **b)** False  
- **c)** False  
- **d)** True  
- **e)** False  
- **f)** True  
- **g)** True  
- **h)** True  
- **i)** True  
- **j)** False  

Would you like further details on any of the statements or concepts?

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**Related questions:**
1. What is the difference between \( \in \) and \( \subseteq \)?
2. How do we determine the cardinality of a set?
3. What is a power set and how is it calculated?
4. Can a set be an element of another set, and how does that work?
5. How do we calculate the number of elements in a power set?

**Tip:** Always remember that numbers (like 1 or 2) cannot be subsets; only sets can be subsets of other sets.

Sea A = {∅, {1, 2}, {1}, {∅}, 1, {2}}. ¿Cuáles de las siguientes afirmaciones son ciertas y por qué? a) 1 ∈ A, b) 2 ∈ A, c) 2 ⊆ A, d) {2} ∈ A, e) {2} ⊆ A, f) ∅ ⊆ A y ∅ ∈ A, g) 1 ∈ A y {1} ∈ A, h) η(A) = 6, i) η(P(A)) = 64, j) {1, 2} ⊆ A.

Analyze and determine the truth of multiple statements regarding set A = {∅, {1, 2}, {1}, {∅}, 1, {2}}. Learn about set elements, subsets, cardinality, and power sets with this step-by-step solution. Suitable for high school students in Grades 9-12.

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