2.7 Partial Orders and Equivalence

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

     equivalence relation partial but not a total order total order none of the above equivalence relation 



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

   
   
   Exercise 2

     equivalence relation partial but not a total order total order none of the above total order 
   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

     equivalence relation partial but not a total order total order none of the above none of the above 
   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.
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