If A = {4, 8} and B ={p, q, r}, then `A xx B` is equal to

Options

A. {(4, p), (8, q), (4, q)}
B. {(4, p) , (4r), (8, p), (8, r)}
C. {(4p, 8q), (4r, 8q)}
D. {(4, p), (4, q), (4, r), (8, p), (8, q), (8, r)}

Answer

To find the Cartesian product \( A \times B \) where \( A = \{4, 8\} \) and \( B = \{p, q, r\} \), we will follow these steps:

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

### Step 2: List the Elements of Set A and Set B
- Set \( A \) has elements: \( 4, 8 \)
- Set \( B \) has elements: \( p, q, r \)

### Step 3: Form Ordered Pairs
Now, we will form ordered pairs by taking each element from set \( A \) and pairing it with each element from set \( B \):

1. Pair the first element of \( A \) (which is \( 4 \)) with each element of \( B \):
- \( (4, p) \)
- \( (4, q) \)
- \( (4, r) \)

2. Pair the second element of \( A \) (which is \( 8 \)) with each element of \( B \):
- \( (8, p) \)
- \( (8, q) \)
- \( (8, r) \)

### Step 4: Combine All Ordered Pairs
Now we will combine all the ordered pairs we formed:
- From \( 4 \): \( (4, p), (4, q), (4, r) \)
- From \( 8 \): \( (8, p), (8, q), (8, r) \)

Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{ (4, p), (4, q), (4, r), (8, p), (8, q), (8, r) \} \]

### Step 5: Count the Number of Elements
The total number of ordered pairs formed is \( 2 \) (from set \( A \)) multiplied by \( 3 \) (from set \( B \)), which gives us \( 2 \times 3 = 6 \) elements in the Cartesian product.

### Final Answer
Therefore, \( A \times B = \{ (4, p), (4, q), (4, r), (8, p), (8, q), (8, r) \} \).

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