Question

Let \( A = \{2, 3\} \) and \( B = \{5, 6\} \), then the number of relations from \( A \times B \) to \( A \times B \) are:

• A. \( 2^{12} \)
• B. \( 2^{14} \)
• C. \( 2^{16} \)
• D. \( 2^{18} \)

Hint:

Don't confuse "number of elements in the Cartesian product" with the "number of relations." The former is \( n \times m \), the latter is \( 2^{n \times m} \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A relation from set \( X \) to set \( Y \) is a subset of the Cartesian product \( X \times Y \). If \( n(X) = p \) and \( n(Y) = q \), then the total number of relations is \( 2^{pq} \).

Step 2: Key Formula or Approach:
1. Calculate \( n(A \times B) \). 2. The number of relations from \( S \) to \( S \) is \( 2^{n(S) \times n(S)} \).

Step 3: Detailed Explanation:
1. \( n(A) = 2, n(B) = 2 \). 2. \( n(A \times B) = 2 \times 2 = 4 \). 3. Let \( S = A \times B \). We need the number of relations from \( S \) to \( S \). 4. Number of relations \( = 2^{n(S) \cdot n(S)} = 2^{4 \cdot 4} = 2^{16} \).

Step 4: Final Answer:
The number of relations is \( 2^{16} \).