Question:
Construct the composition table for \( \times_4 \) on the set \( S = \{0, 1, 2, 3\} \).
Concept:
The operation \( a \times_4 b \) means multiplication modulo 4, i.e.,
\[ a \times_4 b = (a \times b) \mod 4 \]
Solution:
Step 1: Compute each entry using modulo 4.
Examples:
- \( 2 \times_4 3 = 6 \mod 4 = 2 \)
- \( 3 \times_4 3 = 9 \mod 4 = 1 \)
- \( 1 \times_4 2 = 2 \)
Step 2: Fill the table
\[ \begin{array}{c|cccc} \times_4 & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 0 & 2 \\ 3 & 0 & 3 & 2 & 1 \\ \end{array} \]
Final Answer:
The above table is the required composition table for multiplication modulo 4 on the set \( S = \{0,1,2,3\} \).