A multiplication table is an array showing the result of applying a binary operator to elements of a given set S. For example, the following table is the multiplication table for ordinary multiplication.
 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
The results of any binary mathematical operation can be written as a multiplication table. For example, groups have multiplication tables, where the group operation is understood as multiplication. However, different labelings and orderings of a multiplication table may describe the same abstract group. For example, the multiplication table for the cyclic group C4 may be written in three equivalent ways--denoted here by 
,
,
--by permuting the symbols used for the group elements (Cotton 1990, p. 11).
The first such table can be written as follow.
|
1 | A | B | C |
| 1 | 1 | A | B | C |
| A | A | B | C | 1 |
| B | B | C | 1 | A |
| C | C | 1 | A | B |
The multiplication table for a second representation may be obtained from by interchanging A and B.
|
1 | A | B | C |
| 1 | 1 | A | B | C |
| A | A | 1 | C | B |
| B | B | C | A | 1 |
| C | C | B | 1 | A |
And finally, a multiplication table for the third representation can be obtained from by interchanging A and C.
|
1 | A | B | C |
| 1 | 1 | A | B | C |
| A | A | C | 1 | B |
| B | B | 1 | C | A |
| C | C | B | A | 1 |
Abstract Group, Binary Operator, Group, Truth Table
