Step 1: First, determine the sets \( A \) and \( B \):
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:
\[ 2^9 \]
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.