Construct the composition table for \[{ \times _4}\] on set S = {0, 1, 2, 3}.
ANSWER
Verified
Hint: A composition table is a result of operations on the elements of the set. The operation \[{ \times _4}\] gives the remainder of the product of the operands.
Complete step-by-step answer:
A binary operation in a finite set of elements can be completely described by the composition table.
As a first step, we write the elements of the set in the first row and the first column. Each box in the table represents an ordered pair.
Now, the operation \[{ \times _4}\] gives the remainder of the product of the operands. Hence, we find the value of the operator \[{ \times _4}\] on all the ordered pairs and then fill the table.
First, we observe that the operator \[{ \times _4}\] is commutative, hence, the order of elements does not matter. So, we have the following possibilities:
\[{ \times _4}\] acting on (0, 0), we get as follows:
\[0{ \times _4}0 = (0 \times 0)\% 4\]
The percentage symbol stands for the remainder of the expression when divided by the second term.
When 0 is divided by 4, the remainder is 0, hence, we have:
\[0{ \times _4}0 = 0\]
\[{ \times _4}\] acting on (0, 1), we get as follows:
\[0{ \times _4}1 = (0 \times 1)\% 4 = 0\]
Similarly for other terms, we have as follows:
\[0{ \times _4}2 = (0 \times 2)\% 4 = 0\]
\[0{ \times _4}3 = (0 \times 3)\% 4 = 0\]
\[0{ \times _4}4 = (0 \times 4)\% 4 = 0\]
\[1{ \times _4}1 = (1 \times 1)\% 4 = 1\]
\[1{ \times _4}2 = (1 \times 2)\% 4 = 2\]
\[1{ \times _4}3 = (1 \times 3)\% 4 = 3\]
\[1{ \times _4}4 = (1 \times 4)\% 4 = 0\]
\[2{ \times _4}2 = (2 \times 2)\% 4 = 0\]
\[2{ \times _4}3 = (2 \times 6)\% 4 = 2\]
\[2{ \times _4}4 = (2 \times 4)\% 4 = 0\]
\[3{ \times _4}3 = (3 \times 3)\% 4 = 1\]
\[3{ \times _4}4 = (3 \times 4)\% 4 = 0\]
\[4{ \times _4}4 = (4 \times 4)\% 4 = 0\]
Now, we substitute the values in the table to get as follows:
\[{ \times _4}\]
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
0
2
0
2
0
2
0
3
0
3
2
1
0
4
0
0
0
0
0
Hence, we constructed the composition table.
Note: You need not compute all the values individually. You know that 0 multiplied with any number is 0 and hence, the remainder is 0. Also, 4 multiplied with any number is a multiple of 4 and hence, the remainder is again 0.