If `A={1,2,3},B={4,5},C={1,2,3,4,5}` find `(i)AxxB (ii) CxxB (iii) BxxB` Hence prove that `(CxxB)-(AxxB)=BxxB`.

Answer

To solve the problem step by step, we need to find the Cartesian products of the sets \( A \), \( B \), and \( C \), and then prove that \( (C \times B) - (A \times B) = B \times B \).

### Step 1: Find \( A \times B \)

Given:
- \( A = \{1, 2, 3\} \)
- \( B = \{4, 5\} \)

The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

Calculating \( A \times B \):
- For \( a = 1 \): \( (1, 4), (1, 5) \)
- For \( a = 2 \): \( (2, 4), (2, 5) \)
- For \( a = 3 \): \( (3, 4), (3, 5) \)

Thus,
\[ A \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\} \]

### Step 2: Find \( C \times B \)

Given:
- \( C = \{1, 2, 3, 4, 5\} \)

Calculating \( C \times B \):
- For \( c = 1 \): \( (1, 4), (1, 5) \)
- For \( c = 2 \): \( (2, 4), (2, 5) \)
- For \( c = 3 \): \( (3, 4), (3, 5) \)
- For \( c = 4 \): \( (4, 4), (4, 5) \)
- For \( c = 5 \): \( (5, 4), (5, 5) \)

Thus,
\[ C \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5), (5, 4), (5, 5)\} \]

### Step 3: Find \( B \times B \)

Calculating \( B \times B \):
- For \( b_1 = 4 \): \( (4, 4), (4, 5) \)
- For \( b_2 = 5 \): \( (5, 4), (5, 5) \)

Thus,
\[ B \times B = \{(4, 4), (4, 5), (5, 4), (5, 5)\} \]

### Step 4: Prove that \( (C \times B) - (A \times B) = B \times B \)

Now we need to subtract \( A \times B \) from \( C \times B \):

1. \( C \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5), (5, 4), (5, 5)\} \)
2. \( A \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\} \)

Now, we will remove all elements of \( A \times B \) from \( C \times B \):
- Remove \( (1, 4) \)
- Remove \( (1, 5) \)
- Remove \( (2, 4) \)
- Remove \( (2, 5) \)
- Remove \( (3, 4) \)
- Remove \( (3, 5) \)

After removing these pairs, we are left with:
\[ (C \times B) - (A \times B) = \{(4, 4), (4, 5), (5, 4), (5, 5)\} \]

This is exactly \( B \times B \):
\[ B \times B = \{(4, 4), (4, 5), (5, 4), (5, 5)\} \]

### Conclusion

Thus, we have proved that:
\[ (C \times B) - (A \times B) = B \times B \]