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 \(S\) to set \(T\) is any subset of the Cartesian product \(S \times T\).
The total number of relations from \(S\) to \(T\) is \(2^{n(S) \cdot n(T)}\).
Step 3: Detailed Explanation:
First, calculate the set \(A \times B\):
\(A = \{2, 3\} \Rightarrow n(A) = 2\)
\(B = \{5, 6\} \Rightarrow n(B) = 2\)
Then, \(n(A \times B) = n(A) \times n(B) = 2 \times 2 = 4\).
We are looking for relations from \(S\) to \(S\), where \(S = A \times B\).
Total relations \( = 2^{n(S) \cdot n(S)} = 2^{4 \cdot 4} = 2^{16}\).
Step 4: Final Answer:
The total number of relations is \(2^{16}\).