Question

Construct the composition table for ${ \times _{\text{6}}}$on the set S = {0, 1, 2, 3, 4, 5}

Answer 1

Hint: To construct the composition table for ${ \times _{\text{6}}}$ on the given set S, we should first understand how a composition table is defined and how it is constructed. In a composition table on a set with 6 elements as given above, if the product of numbers is equal to or less than five, we write the value and if it is above we write the remainder of the product divided by 6.

Complete step by step answer:
Given Data,
Composition table for ${ \times _{\text{6}}}$on the set S = {0, 1, 2, 3, 4, 5}

Now let us consider a and b to be two numbers which belong to the set ‘S’.
Now the composition table for ${ \times _{\text{6}}}$on the set S is defined as:
${\text{a }}{ \times _{\text{6}}}{\text{b = }}\left\{
  {\text{a}}{\text{.b if a}} \leqslant 5 \\
  {\text{Remainder of }}\dfrac{{{\text{a}}{\text{.b}}}}{6}{\text{ if a}}{\text{.b > 5}} \\
  \right\}$

Now we perform this operation on each element of the set with every element of the set itself.
We express it in the form of a 6 × 6 table, with elements from 0 to 6 in the first rows and first column sequentially.
It looks as follows:
${\text{a }}{ \times _{\text{6}}}{\text{b = }}\boxed{\begin{array}{*{20}{c}}
  {{ \times _{\text{6}}}}&{\left( 0 \right)}&{\left( 1 \right)}&{\left( 2 \right)}&{\left( 3 \right)}&{\left( 4 \right)}&{\left( 5 \right)} \\
  {\left( 0 \right)}&0&0&0&0&0&0 \\
  {\left( 1 \right)}&0&1&2&3&4&5 \\
  {\left( 2 \right)}&0&2&4&0&2&4 \\
  {\left( 3 \right)}&0&3&0&3&0&3 \\
  {\left( 4 \right)}&0&4&2&0&4&2 \\
  {\left( 5 \right)}&0&5&4&3&2&1
\end{array}}$
This is called the composition table for ${ \times _{\text{6}}}$on the set S = {0, 1, 2, 3, 4, 5}.

Note: In order to solve this type of problems the key is to know the concept of a composition table for a given set and the procedure to perform it. It can be of different types based on the number of elements in the set and the value of elements in the set. The order of elements mentioned inside the set is to be considered. We perform the calculation for each position of the table and fill it in an order to obtain the final output.