Consider the following
1. `A cup (B cap C) = (A cap B) cup (A cap C)`
II. `A cap (B cup C) = (A cup B) cap (A cup C)`
Which of the above statement(s) is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements given:

1. **Statement I:** \( A \cup (B \cap C) = (A \cap B) \cup (A \cap C) \)
2. **Statement II:** \( A \cap (B \cup C) = (A \cup B) \cap (A \cup C) \)

We will check the validity of each statement step by step.

### Step 1: Analyze Statement I
We need to check if \( A \cup (B \cap C) = (A \cap B) \cup (A \cap C) \).

**Left Side:**
- \( A \cup (B \cap C) \) means we take all elements in set A and all elements that are in both B and C.

**Right Side:**
- \( (A \cap B) \cup (A \cap C) \) means we take all elements that are both in A and B, and all elements that are both in A and C.

To see if these two sides are equal, we can use a Venn diagram or logical reasoning:
- The left side includes all elements from A and any elements that are common to both B and C.
- The right side includes elements that are in A and also those that are in both B and C.

After analyzing, we can see that these two expressions are not equal. Thus, **Statement I is incorrect.**

### Step 2: Analyze Statement II
Next, we check if \( A \cap (B \cup C) = (A \cup B) \cap (A \cup C) \).

**Left Side:**
- \( A \cap (B \cup C) \) means we take all elements that are in A and also in either B or C.

**Right Side:**
- \( (A \cup B) \cap (A \cup C) \) means we take elements that are in either A or B and also in either A or C.

Again, we can analyze:
- The left side includes elements that are in A and also in either B or C.
- The right side includes elements that are in A, or in B, and also in A, or in C.

Through logical reasoning or using a Venn diagram, we can show that these two expressions are indeed equal. Thus, **Statement II is correct.**

### Conclusion
- **Statement I is incorrect.**
- **Statement II is correct.**

Therefore, the answer to the question is that only the second statement is correct.

### Final Answer
Only Statement II is correct.