# Interactive Real Analysis

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## 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:
1. r : { (a,a), (b,b), (a,b), (b,a) } (on B)
2. s : 1 ~ 1, 2 ~ 2, 3 ~ 3, 4 ~ 4, 1 ~ 4, 4 ~ 1, 2 ~ 4, 4 ~ 2 (on A)
Which one is an equivalence relation, if any ?
1. 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.

2. 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.
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