If A and B are any two sets, then

Options

A. `A xx B = B xx A`
B. `A xx B sub B xx A`
C. `B xx A sub A xx B`
D. `n(A xx B) = n(B xx A)`

Answer

To solve the problem step by step, we need to analyze the properties of the Cartesian product of two sets A and B.

### Step 1: Define the Sets
Let's assume:
- Set A = {1, 2}
- Set B = {3, 4}

### Step 2: Find A × B
The Cartesian product A × B is defined as the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Calculating A × B:
- A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}

### Step 3: Find B × A
Now, let's find the Cartesian product B × A, which is the set of all ordered pairs (b, a) where b ∈ B and a ∈ A.

Calculating B × A:
- B × A = {(3, 1), (3, 2), (4, 1), (4, 2)}

### Step 4: Compare A × B and B × A
Now we need to compare A × B and B × A:
- A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}
- B × A = {(3, 1), (3, 2), (4, 1), (4, 2)}

Clearly, A × B is not equal to B × A because the ordered pairs are different.

### Step 5: Check the Options
Now we check the given options:
1. A × B = B × C (not true, as shown)
2. A × B is a subset of B × A (not true, as shown)
3. The number of elements in A × B is equal to the number of elements in B × A (true, both have 4 elements).

### Conclusion
The correct option is that the number of elements in A × B is equal to the number of elements in B × A.