If `A={1,3,5,7)` and `B={2,4,6}`, find `AxxB`

Answer

To find the Cartesian product \( A \times B \) of the sets \( A = \{1, 3, 5, 7\} \) and \( B = \{2, 4, 6\} \), we will follow these steps:

### Step 1: Understand the Definition
The Cartesian product \( A \times B \) consists of all possible ordered pairs \( (a, b) \) where \( a \) is an element from set \( A \) and \( b \) is an element from set \( B \).

### Step 2: List Elements of Set A
The elements of set \( A \) are:
- \( 1 \)
- \( 3 \)
- \( 5 \)
- \( 7 \)

### Step 3: List Elements of Set B
The elements of set \( B \) are:
- \( 2 \)
- \( 4 \)
- \( 6 \)

### Step 4: Create Ordered Pairs
Now, we will create ordered pairs by taking each element from set \( A \) and pairing it with each element from set \( B \).

1. For \( a = 1 \):
- Pair with \( 2 \): \( (1, 2) \)
- Pair with \( 4 \): \( (1, 4) \)
- Pair with \( 6 \): \( (1, 6) \)

2. For \( a = 3 \):
- Pair with \( 2 \): \( (3, 2) \)
- Pair with \( 4 \): \( (3, 4) \)
- Pair with \( 6 \): \( (3, 6) \)

3. For \( a = 5 \):
- Pair with \( 2 \): \( (5, 2) \)
- Pair with \( 4 \): \( (5, 4) \)
- Pair with \( 6 \): \( (5, 6) \)

4. For \( a = 7 \):
- Pair with \( 2 \): \( (7, 2) \)
- Pair with \( 4 \): \( (7, 4) \)
- Pair with \( 6 \): \( (7, 6) \)

### Step 5: Compile the Result
Now, we can compile all the ordered pairs we created:
\[ A \times B = \{ (1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6), (7, 2), (7, 4), (7, 6) \} \]

### Final Answer
Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{ (1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6), (7, 2), (7, 4), (7, 6) \} \]