Question:
Construct the composition table for \( \times_6 \) on the set \( S = \{0,1,2,3,4,5\} \).
Concept:
The operation \( a \times_6 b \) means multiplication modulo 6:
\[ a \times_6 b = (a \times b) \mod 6 \]
Solution:
Step 1: Multiply the numbers and take remainder after division by 6.
- \( 2 \times_6 3 = 6 \mod 6 = 0 \)
- \( 4 \times_6 5 = 20 \mod 6 = 2 \)
- \( 3 \times_6 4 = 12 \mod 6 = 0 \)
Step 2: Construct the table
\[ \begin{array}{c|cccccc} \times_6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \\ \end{array} \]
Final Answer:
The above table is the required composition table for multiplication modulo 6 on the set \( S = \{0,1,2,3,4,5\} \).