Question

Construct the composition table for \[{ \times _4}\] on set S = {0, 1, 2, 3}.

Answer 1

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.