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.

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

\(\{(a,b) \;\; a \text{ is the age of someone who is not younger than anyone of age } b\}\)
Ages can be translated into days or similar numerical units, which reveals that we have just given a somewhat awkward description of the relation greaterorequal on these numbers. 
\(\{(p,q)  p \text{ is a person whose age is an integer multiple of person \(q\)'s age}\}\)
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 4yearold is related to a 2yearold, 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.