Suppose A, B and C are three sets. Consider the following:
(i) `A cup (B cap C) = (A cup B) cap (A cup C)`
(ii) `(A cap B) cup C = (A cup C) cap (B cup C)`
Which of the above is true?

Options

A. (i)only
B. (ii)only
C. Both (i) and (ii)
D. Neither of (i) and (ii)

Answer

To determine the validity of the statements regarding the sets A, B, and C, we will analyze each statement step by step.

### Step 1: Analyze Statement (i)
The statement is:
\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \]

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

**Right Side:**
- \( (A \cup B) \cap (A \cup C) \) means we first take all elements from A and B, and then all elements from A and C. The intersection will give us the elements that are common in both results.

### Step 2: Use Venn Diagrams
To visualize this, we can draw a Venn diagram with three overlapping circles representing sets A, B, and C.

1. The left side \( A \cup (B \cap C) \) will include:
- All elements in A.
- Plus the elements that are in both B and C.

2. The right side \( (A \cup B) \cap (A \cup C) \) will include:
- All elements in A.
- Plus the elements that are in both B and C, but only those that also appear in A.

### Step 3: Compare Both Sides
From the Venn diagram, we can see that both sides represent the same set of elements:
- Both sides include all elements of A.
- Both sides include the elements that are common to B and C.

Thus, **Statement (i) is true**.

### Step 4: Analyze Statement (ii)
The statement is:
\[ (A \cap B) \cup C = (A \cup C) \cap (B \cup C) \]

**Left Side:**
- \( (A \cap B) \cup C \) means we take all elements that are common to both A and B, and then add all elements from C.

**Right Side:**
- \( (A \cup C) \cap (B \cup C) \) means we first take all elements from A and C, and then all elements from B and C. The intersection will give us the elements that are common in both results.

### Step 5: Use Venn Diagrams Again
1. The left side \( (A \cap B) \cup C \) will include:
- All elements that are in both A and B.
- Plus all elements in C.

2. The right side \( (A \cup C) \cap (B \cup C) \) will include:
- All elements from A and C.
- All elements from B and C, but only those that also appear in both.

### Step 6: Compare Both Sides
From the Venn diagram, we can see that both sides represent the same set of elements:
- Both sides include all elements from C.
- Both sides include the elements that are common to A and B.

Thus, **Statement (ii) is also true**.

### Conclusion
Both statements (i) and (ii) are true.