## 1.3. Equivalence Relations and Classes

### Examples 1.3.2:

Let A = {1, 2, 3, 4} and B = {a, b, c} and define the following two
relations:

*r : { (a,a), (b,b), (a,b), (b,a) }*(on B)*s*: 1 ~ 1, 2 ~ 2, 3 ~ 3, 4 ~ 4, 1 ~ 4, 4 ~ 1, 2 ~ 4, 4 ~ 2 (on A)

- Consider the relation
*r: { (a,a), (b,b), (a,b), (b,a) }*. Then this is, technically speaking,**not**an equivalence relation, for the simple reason that the element c has no association. However, if we restrict our relation to a new domain that does not include the element*c*, then it might be an equivalence relation. We need to check:**reflexive**: every element that has a relation does also have a relation with itself**symmetric**: clearly true**transitive**: follows from the first two in this case.

- Consider the relation
*s*: 1 ~ 1, 2 ~ 2, 3 ~ 3, 4 ~ 4, 1 ~ 4, 4 ~ 1, 2 ~ 4, 4 ~ 2. Then this is**not**an equivalence relation. We need to check:**reflexive**: every element with a relation is related to itself.**symmetric**: true as well (3 is only related to 3, so symmetry condition does not apply here)**not transitive**: 1 ~ 4 and 4 ~ 2, but 1 is not related to 2.