Mark the true statements. The power set of the power set of the empty set is
{∅,∅}
.
{1,2}
is a subset of the power set of
{1,2,3}
. The power set of the empty set is the empty set. The power set of the power set of the empty set is
{{∅,{∅}}}
. The power set of the power set of the empty set is
{∅,{∅}}
.
{{1,2}}
is a subset of the power set of
{1,2,3}
. The set
{{}}
is not empty. The power set of the empty set is
{∅}
.
{1,2}
is an element of the power set of
{1,2,3}
.
{{1,2}}
is an element of the power set of
{1,2,3}
.

Let’s break down the given statements one by one and assess their truth:

### 1. **The power set of the power set of the empty set is {∅, ∅}.**
   - **False.** The power set of the empty set ∅ is P(∅) = {∅}. The power set of P(∅) would be P({∅}) = {∅, {∅}}. The set {∅, ∅} is not a valid set (repetition of the same element), and the correct power set is {∅, {∅}}.

### 2. **{1, 2} is a subset of the power set of {1, 2, 3}.**
   - **False.** The power set of {1, 2, 3} contains subsets of {1, 2, 3}, but {1, 2} is a set of elements, not a set of subsets. For this to be true, the elements {1, 2} would need to be sets themselves, not just numbers.

### 3. **The power set of the empty set is the empty set.**
   - **False.** The power set of the empty set is {∅}, not ∅. The power set of any set always contains at least the empty set as an element.

### 4. **The power set of the power set of the empty set is {{∅, {∅}}}.**
   - **False.** The power set of P(∅) is {∅, {∅}}, but {{∅, {∅}}} would imply a different structure. The correct power set of P(∅) is {∅, {∅}}, not {{∅, {∅}}}.

### 5. **The power set of the power set of the empty set is {∅, {∅}}.**
   - **True.** As explained, the power set of P(∅) is indeed {∅, {∅}}.

### 6. **{{1, 2}} is a subset of the power set of {1, 2, 3}.**
   - **True.** The set {1, 2} is a subset of {1, 2, 3}, so {{1, 2}} (a set containing the subset {1, 2}) is a subset of P({1, 2, 3}).

### 7. **The set {{}} is not empty.**
   - **True.** The set {{}} is not empty because it contains one element, the empty set.

### 8. **The power set of the empty set is {∅}.**
   - **True.** The power set of ∅ is {∅}, which contains exactly one element: the empty set.

### 9. **{1, 2} is an element of the power set of {1, 2, 3}.**
   - **True.** {1, 2} is a subset of {1, 2, 3}, so it is an element of P({1, 2, 3}).

### 10. **{{1, 2}} is an element of the power set of {1, 2, 3}.**
   - **False.** {{1, 2}} is a set containing a subset {1, 2}, but {1, 2} itself is an element of P({1, 2, 3}), not {{1, 2}}.

---

### Summary of True Statements:
- The power set of the power set of the empty set is {∅, {∅}}.
- {{1, 2}} is a subset of the power set of {1, 2, 3}.
- The set {{}} is not empty.
- The power set of the empty set is {∅}.
- {1, 2} is an element of the power set of {1, 2, 3}.

Would you like further clarification or more details on any of these statements?

---

**Related Questions:**
1. What is the power set of a set containing three elements?
2. How do subsets differ from elements in power sets?
3. Can a power set ever contain an infinite number of elements?
4. How is the size of a power set determined?
5. What is the power set of a set containing another set?

**Tip:** The size of a power set is always 2 raised to the power of the number of elements in the original set (e.g., for a set with n elements, its power set has \(2^n\) elements).

Mark the true statements. The power set of the power set of the empty set is {∅,∅}. {1,2} is a subset of the power set of {1,2,3}. The power set of the empty set is the empty set. The power set of the power set of the empty set is {{∅,{∅}}}. The power set of the power set of the empty set is {∅,{∅}}. {{1,2}} is a subset of the power set of {1,2,3}. The set {{}} is not empty. The power set of the empty set is {∅}. {1,2} is an element of the power set of {1,2,3}. {{1,2}} is an element of the power set of {1,2,3}.

Explore the truth behind statements involving power sets, subsets, and the empty set. Learn how to determine whether {1, 2} is a subset of the power set of {1, 2, 3}, and uncover the true structure of power sets like P(P(∅)). Suitable for university-level discrete mathematics students.

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