Question

If \( A \) is the set of even natural numbers less than 8 and \( B \) is the set of prime numbers less than 7, then the number of relations from \( A \) to \( B \) is:

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

Hint:

The number of relations between two sets \( A \) and \( B \) is \( 2^{|A \times B|} \), where \( |A \times B| \) is the number of ordered pairs in the Cartesian product of \( A \) and \( B \).

Answer 1

The Correct Option is A

We are given two sets: - \( A \) is the set of even natural numbers less than 8. Therefore, \( A = \{2, 4, 6\} \). - \( B \) is the set of prime numbers less than 7. Therefore, \( B = \{2, 3, 5\} \). A relation from set \( A \) to set \( B \) is any subset of the Cartesian product \( A \times B \), which consists of all possible ordered pairs of elements from \( A \) and \( B \). The number of elements in \( A \times B \) is the product of the number of elements in \( A \) and \( B \): \[ |A \times B| = |A| \times |B| = 3 \times 3 = 9 \] The number of relations from \( A \) to \( B \) is the number of subsets of \( A \times B \), which is \( 2^{|A \times B|} \). Since \( |A \times B| = 9 \), the number of relations is: \[ 2^9 \] Thus, the correct answer is \( \boxed{2^9} \).