For non-empty sets A, B and C, the following two statements are given:
Statement P: `A nn (B uu C)=(A nn B)uu C`
Statement Q: Cis a subset of A
Which one of the following is correct?

Options

A. `P lArr Q`
B. `P hArr Q`
C. `P rArr Q`
D. Nothing can be said about the correctness of the above three with certainty

Answer

To solve the problem, we need to analyze the two statements given about the sets A, B, and C.

### Step 1: Understand Statement P
Statement P states that:
\[ A \cap (B \cup C) = (A \cap B) \cup C \]

To check if this statement is true, we can use the distributive property of set operations. The distributive property states that:
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]

### Step 2: Compare Statement P with the Distributive Property
From the distributive property, we have:
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]

Now, if we compare this with Statement P:
- Left side: \( A \cap (B \cup C) \)
- Right side: \( (A \cap B) \cup C \)

For these two expressions to be equal, \( (A \cap C) \) must equal \( C \). This is only true if \( C \) is a subset of \( A \).

### Step 3: Understand Statement Q
Statement Q states that:
\[ C \subseteq A \]

This means that every element of set C is also an element of set A.

### Step 4: Conclusion
From our analysis:
- Statement P holds true if and only if Statement Q is true, meaning \( C \) must be a subset of \( A \).

Thus, the correct conclusion is that Statement Q is correct.

### Final Answer
The correct statement is:
**Statement Q: C is a subset of A.**

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