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

Answer

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

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

### Step 2: List the Elements of Set B and Set A
- Set \( B = \{2, 4, 6\} \)
- Set \( A = \{1, 3, 5, 7\} \)

### Step 3: Create Ordered Pairs
We will pair each element of set \( B \) with every element of set \( A \):

1. For \( b = 2 \):
- Pair with \( a = 1 \): \( (2, 1) \)
- Pair with \( a = 3 \): \( (2, 3) \)
- Pair with \( a = 5 \): \( (2, 5) \)
- Pair with \( a = 7 \): \( (2, 7) \)

2. For \( b = 4 \):
- Pair with \( a = 1 \): \( (4, 1) \)
- Pair with \( a = 3 \): \( (4, 3) \)
- Pair with \( a = 5 \): \( (4, 5) \)
- Pair with \( a = 7 \): \( (4, 7) \)

3. For \( b = 6 \):
- Pair with \( a = 1 \): \( (6, 1) \)
- Pair with \( a = 3 \): \( (6, 3) \)
- Pair with \( a = 5 \): \( (6, 5) \)
- Pair with \( a = 7 \): \( (6, 7) \)

### Step 4: Compile All Ordered Pairs
Now, we will compile all the ordered pairs we created:

\[ B \times A = \{(2, 1), (2, 3), (2, 5), (2, 7), (4, 1), (4, 3), (4, 5), (4, 7), (6, 1), (6, 3), (6, 5), (6, 7)\} \]

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