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. \( 2^{16} \)
• C. \( 2^{10} \)
• D. \( 2^{15} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A relation from set $X$ to set $Y$ is any subset of the Cartesian product $X \times Y$. If $|X| = m$ and $|Y| = n$, the total number of relations is $2^{m \times n}$.
Step 2: Key Formula or Approach:
Number of relations = $2^{|X| \cdot |Y|}$. In this case, both $X$ and $Y$ are the set $A \times B$.
Step 3: Detailed Explanation:
1. First, find the number of elements in $A \times B$: \[ |A| = 2, |B| = 2 \implies |A \times B| = 2 \times 2 = 4 \] 2. Let $S = A \times B$. We are looking for the number of relations from $S$ to $S$. 3. The number of elements in $S \times S$ is: \[ |S| \times |S| = 4 \times 4 = 16 \] 4. The number of relations is the number of subsets of $S \times S$, which is $2^{16}$.
Step 4: Final Answer:
The number of relations is \( 2^{16} \).