 QUESTION

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

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.

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.