Question

If \( A = \{3, 5, 7\} \) and \( B = \{1, 2, 3, 5\} \), then \( A \times B \cap B \times A \) is equal to:

• A. $\{(3, 3), (5, 3), (2, 7), (7, 2)\}$
• B. $\{(3, 3), (3, 5), (5, 3), (5, 5)\}$
• C. $\{(3, 3), (5, 5)\}$
• D. $\{(3, 5), (5, 5), (5, 3)\}$

Hint:

For $A \times B \cap B \times A$, only pairs where both elements are in $A \cap B$ are valid.

Answer 1

The Correct Option is B

We are asked to find the intersection of the Cartesian products \( A \times B \) and \( B \times A \), where:

• \( A = \{3, 5, 7\} \), and
• \( B = \{1, 2, 3, 5\} \).

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

The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) such that \( a \in A \) and \( b \in B \). Thus, \[ A \times B = \{ (3, 1), (3, 2), (3, 3), (3, 5), (5, 1), (5, 2), (5, 3), (5, 5), (7, 1), (7, 2), (7, 3), (7, 5) \}. \]

Step 2: Find \( B \times A \)

Similarly, the Cartesian product \( B \times A \) consists of all ordered pairs \( (b, a) \) such that \( b \in B \) and \( a \in A \). Thus, \[ B \times A = \{ (1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7) \}. \]

Step 3: Find \( A \times B \cap B \times A \)

The intersection of \( A \times B \) and \( B \times A \) consists of pairs that appear in both sets. To find the common pairs, we compare the elements of \( A \times B \) and \( B \times A \): - From \( A \times B \), we have the pairs \( (3, 1), (3, 2), (3, 3), (3, 5), (5, 1), (5, 2), (5, 3), (5, 5), (7, 1), (7, 2), (7, 3), (7, 5) \). - From \( B \times A \), we have the pairs \( (1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7) \). By comparing, we see the common pairs are: \[ (3, 3), (5, 5) \]

Final Answer:

The intersection \( A \times B \cap B \times A \) is equal to \( \{ (3, 3), (5, 5) \} \).