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}}.

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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?

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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).