Composition Table or Cayley Table
A binary operation in a finite set can completely be described by means of a table. This table is known as a composition table. The composition table helps us to verify most of the properties satisfied by the binary operations. This table can be formed as follows:
(i) Write the elements of the set (which are finite in number) in a row as well as in a column.
(ii) Write the element associated to the ordered pair $$\left( {{a_i},{a_j}} \right)$$ at the intersection of the row headed by $${a_i}$$ and the column headed by $${a_j}$$. Thus ($$ith$$ entry on the left) $$ \times $$($$jth$$ entry on the top) = entry where the $$ith$$ row and $$jth$$ column intersect.
For example, the composition table for the group $$\left\{ {0,1,2,3,4} \right\}$$ for the operation of addition is given below:
0 | 1 | 2 | 3 | 4 | |
0 | 0 | 1 | 2 | 3 | 4 |
1 | 1 | 2 | 3 | 4 | 5 |
2 | 2 | 3 | 4 | 5 | 6 |
3 | 3 | 4 | 5 | 6 | 7 |
4 | 4 | 5 | 6 | 7 | 8 |
In the above example, the first element of the first row in the body of the table, 0, is obtained by adding the first element 0 of the head row and the first element 0 of the head column. Similarly the third element of the 4th row (5) is obtained by adding the third element 2 of the head row and the fourth element of the head column and so on.
An operation represented by the composition table will be binary, if every entry of the composition table belongs to the given set. It is to be noted that composition table contains all possible combinations of two elements with respect to the operation.
Note:
(1) It should be noted that the elements of the set should be written in the same order both in the top border and the left border of the table while preparing the composition table.
(2) Generally a table which defines a binary operation “.” on a set is called a multiplication table. When the operation is “+” the table is called an addition table.