Question:
Construct the composition table for \( +_5 \) on the set \( S = \{0,1,2,3,4\} \).
Concept:
The operation \( a +_5 b \) means addition modulo 5, i.e.,
\[ a +_5 b = (a + b) \mod 5 \]
Solution:
Step 1: Add the numbers and take remainder after division by 5.
- \( 2 +_5 3 = 5 \mod 5 = 0 \)
- \( 4 +_5 4 = 8 \mod 5 = 3 \)
- \( 3 +_5 2 = 5 \mod 5 = 0 \)
Step 2: Construct the table
\[ \begin{array}{c|ccccc} +_5 & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 & 4 & 0 \\ 2 & 2 & 3 & 4 & 0 & 1 \\ 3 & 3 & 4 & 0 & 1 & 2 \\ 4 & 4 & 0 & 1 & 2 & 3 \\ \end{array} \]
Final Answer:
The above table is the required composition table for addition modulo 5 on the set \( S = \{0,1,2,3,4\} \).