If `A={a,b},B={2,3} and C={3,4}`, then evaluate:
(i) `Axx(B nn C)" "(ii) Axx(B uu C)`

Explanation

To solve the problem, we need to evaluate two expressions involving the sets \( A \), \( B \), and \( C \).

Given:
- \( A = \{a, b\} \)
- \( B = \{2, 3\} \)
- \( C = \{3, 4\} \)

We will evaluate:
1. \( A \times (B \cap C) \)
2. \( A \times (B \cup C) \)

### Step 1: Evaluate \( B \cap C \)

The intersection of two sets \( B \) and \( C \) includes only the elements that are present in both sets.

- \( B = \{2, 3\} \)
- \( C = \{3, 4\} \)

Now, we find the common elements:
- The element \( 3 \) is present in both \( B \) and \( C \).
- The element \( 2 \) is not in \( C \).
- The element \( 4 \) is not in \( B \).

Thus, the intersection \( B \cap C = \{3\} \).

### Step 2: Evaluate \( A \times (B \cap C) \)

Now we will find the Cartesian product of set \( A \) and the result from Step 1.

- \( A = \{a, b\} \)
- \( B \cap C = \{3\} \)

The Cartesian product \( A \times (B \cap C) \) is formed by pairing each element of \( A \) with each element of \( B \cap C \):
- Pairing \( a \) with \( 3 \) gives \( (a, 3) \).
- Pairing \( b \) with \( 3 \) gives \( (b, 3) \).

Thus, \( A \times (B \cap C) = \{(a, 3), (b, 3)\} \).

### Step 3: Evaluate \( B \cup C \)

The union of two sets \( B \) and \( C \) includes all elements that are in either set.

- \( B = \{2, 3\} \)
- \( C = \{3, 4\} \)

Now, we combine the elements:
- The element \( 2 \) is in \( B \).
- The element \( 3 \) is in both sets but we only include it once.
- The element \( 4 \) is in \( C \).

Thus, the union \( B \cup C = \{2, 3, 4\} \).

### Step 4: Evaluate \( A \times (B \cup C) \)

Now we will find the Cartesian product of set \( A \) and the result from Step 3.

- \( A = \{a, b\} \)
- \( B \cup C = \{2, 3, 4\} \)

The Cartesian product \( A \times (B \cup C) \) is formed by pairing each element of \( A \) with each element of \( B \cup C \):
- Pairing \( a \) with \( 2 \) gives \( (a, 2) \).
- Pairing \( a \) with \( 3 \) gives \( (a, 3) \).
- Pairing \( a \) with \( 4 \) gives \( (a, 4) \).
- Pairing \( b \) with \( 2 \) gives \( (b, 2) \).
- Pairing \( b \) with \( 3 \) gives \( (b, 3) \).
- Pairing \( b \) with \( 4 \) gives \( (b, 4) \).

Thus, \( A \times (B \cup C) = \{(a, 2), (a, 3), (a, 4), (b, 2), (b, 3), (b, 4)\} \).

### Final Answers:
1. \( A \times (B \cap C) = \{(a, 3), (b, 3)\} \)
2. \( A \times (B \cup C) = \{(a, 2), (a, 3), (a, 4), (b, 2), (b, 3), (b, 4)\} \)