Which of the following statements is equivalent to the statement "P(X) UP(Y) = P(XUY)"?
(a) "X = ∅"
(b) "Y = ∅"
(c) "X ⊆ Y"
(d) "X ⊆ Y or Y ⊆ X"
（e） “X∩Y=∅"
（土） “X union Y＝∅"
（g） “X=Y"

The statement in question is:

\[
P(X) \cup P(Y) = P(X \cup Y)
\]

This equation suggests that the union of the power sets of \(X\) and \(Y\) is equal to the power set of the union of \(X\) and \(Y\).

To analyze this, let's first break down what the power set operation and union operation mean:

- \(P(X)\) is the power set of \(X\), meaning the set of all subsets of \(X\).
- \(P(Y)\) is the power set of \(Y\), meaning the set of all subsets of \(Y\).
- \(P(X) \cup P(Y)\) is the set containing all subsets of \(X\) and all subsets of \(Y\).
- \(P(X \cup Y)\) is the set containing all subsets of \(X \cup Y\), the union of \(X\) and \(Y\).

### Examining the Statements:

Let's go through the options to see which condition would make the equation true.

- **(a) \(X = \emptyset\):**
  - If \(X = \emptyset\), then \(P(X) = \{\emptyset\}\). However, this doesn't necessarily simplify \(P(X) \cup P(Y)\) to equal \(P(X \cup Y)\) unless \(Y = \emptyset\) as well. So, this option is not sufficient by itself.

- **(b) \(Y = \emptyset\):**
  - Similar to the above, \(P(Y) = \{\emptyset\}\) when \(Y = \emptyset\). This does not lead to \(P(X) \cup P(Y) = P(X \cup Y)\) unless \(X = \emptyset\) as well. Thus, this is not sufficient by itself.

- **(c) \(X \subseteq Y\):**
  - If \(X \subseteq Y\), then every subset of \(X\) is also a subset of \(Y\). Therefore, \(P(X) \subseteq P(Y)\), and \(P(X) \cup P(Y)\) might still not equal \(P(X \cup Y)\). The subsets of \(Y\) that include elements not in \(X\) would not be in \(P(X)\), so this option does not hold.

- **(d) \(X \subseteq Y\) or \(Y \subseteq X\):**
  - This option means that one set is a subset of the other. In such a case, \(P(X) \cup P(Y)\) might still not equal \(P(X \cup Y)\), because the power set of the union might include subsets that are not present in either \(P(X)\) or \(P(Y)\). Thus, this is not sufficient.

- **(e) \(X \cap Y = \emptyset\):**
  - If \(X\) and \(Y\) are disjoint, \(P(X) \cup P(Y)\) consists of all subsets of \(X\) and \(Y\). However, \(P(X \cup Y)\) would also include subsets that mix elements from both \(X\) and \(Y\), so this option doesn't satisfy the equality.

- **(f) \(X \cup Y = \emptyset\):**
  - If \(X \cup Y = \emptyset\), then both \(X\) and \(Y\) are empty. In this case, \(P(X) = P(Y) = P(\emptyset) = \{\emptyset\}\), and \(P(X) \cup P(Y) = P(X \cup Y) = \{\emptyset\}\). This condition makes the statement true, but it’s too restrictive and doesn't cover all cases.

- **(g) \(X = Y\):**
  - If \(X = Y\), then \(P(X) = P(Y)\) and \(X \cup Y = X = Y\). Therefore, \(P(X) \cup P(Y) = P(X) = P(Y) = P(X \cup Y)\). This option makes the statement always true.

### Conclusion:

The correct option is:

**(g) \(X = Y\)**

This condition ensures that the union of the power sets of \(X\) and \(Y\) equals the power set of the union of \(X\) and \(Y\).

Would you like further details on any specific part of this analysis?

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Here are 5 related questions:

1. What is the relationship between the power sets of two sets when one set is a subset of the other?
2. How does the power set of the union of two sets compare to the union of their power sets?
3. If \(X \cap Y = \emptyset\), how would \(P(X \cap Y)\) compare to \(P(X) \cup P(Y)\)?
4. What are the conditions under which \(P(X \cup Y) = P(X) \cup P(Y)\) for any sets \(X\) and \(Y\)?
5. How would the power set operation differ for finite and infinite sets?

**Tip:** Understanding the properties of power sets can help in solving complex problems involving set operations.

Explore the equivalence statement P(X) ∪ P(Y) = P(X ∪ Y) in set theory. Analyze conditions under which this statement holds true, including detailed explanations and examples. Suitable for advanced mathematics enthusiasts.

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