2.7 Partial Orders and Equivalence

Equivalence Relations & Partial Orders

For each of the following relations, indicate whether it is an equivalence relation, a partial but not a total order, a total order, or none of the above.

1. $$\{(p,q) \;|\; p \text{ and } q \text{ are people of the same age}\}$$

Exercise 1

2. $$\{(a,b) \;|\; a \text{ is the age of someone who is not younger than anyone of age } b\}$$

Exercise 2
Ages can be translated into days or similar numerical units, which reveals that we have just given a somewhat awkward description of the relation greater-or-equal on these numbers.

3. $$\{(p,q) | p \text{ is a person whose age is an integer multiple of person \(q$$'s age}\}\)

Exercise 3
Two different people can be the same age, so the relation is not antisymmetric, ruling out partial order and total order. It is not symmetric, since a 4-year-old is related to a 2-year-old, but not conversely, ruling out equivalence relation. Note that as a relation on their ages, this would be the same as the divisibility relation on nonnegative integers, for which partial but not a total order would have been correct. Yes, this was a bit of a trick question.