Question

Let A be the set of even natural numbers that are<8 and B be the set of prime integers that are<7. The number of relations from A to B is:

• A. \(3^2\)
• B. \(2^9 - 1\)
• C. \(9^2\)
• D. \(2^9\)

Hint:

The number of relations between two sets A and B is equal to the number of subsets of their Cartesian product A × B, which is 2 ^|A×B| .

Answer 1

The Correct Option is D

Step 1: First, determine the sets \( A \) and \( B \):

• \( A = \{2, 4, 6\} \) (even natural numbers less than 8)
• \( B = \{2, 3, 5\} \) (prime integers less than 7)

Step 2: The number of relations from set \( A \) to set \( B \) is given by the total number of subsets of the Cartesian product \( A \times B \). The number of elements in \( A \times B \) is:

\[ |A| \times |B| = 3 \times 3 = 9 \]

Step 3: The number of relations is the number of subsets of \( A \times B \), which is \( 2^9 \), since each pair in \( A \times B \) can either be included in or excluded from the relation. Thus, the correct answer is:

\[ 2^9 \]