Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 1.1. Notation and Set Theory
• 1.2. Relations and Functions
• 1.3. Equivalence Relations and Classes
• 1.4. Natural Numbers, Integers, and Rational Numbers
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 1.3.2:

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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 ?

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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.