This can be dispayed as in Figure 2. A loop on an element means that the element is related to itself (here this occurs for every element as every natural number is a multiple of itself).

- For any two inhabitants and of , we denote if lives next door to . This relation is not reflexive, as a single inhabitant cannot live next door to himself (actually, we should give a counter-example).
- In the same set , we denote if they live in a flat with the same number of rooms. This is a reflexive relation.

- Let be the set of students of an institution. If and are two students in , we write when and come from the same town. The relation is symmetric: for any two students and , saying that `` and come from the same town'' is equivalent (thus implies) saying that `` and come from the same town''.
- Let be the set of children in a kindergarden. For two children and , we write if is the sister of . Being symmetric or not depends on the set itself. If is the sister of and is a boy, then .

- The binary relation of Example 2.5 is not transitive. There we have: and , but .
- Let be the set of all the students in a class. We define the relation on as follows: if and are two students in , we write if and have the same eye color. This relation is transitive: suppose that , and are any three students in such that and have the same eye color and and have the same eye color; then and have the same eye color.

In other words if, and only if, is a multiple of . For instance, and , but . The relation is transitive. Let , and be three positive integers such that and , i.e.

and |

It follows that , and is an integer. Whence the result.