2.7 Partial Orders and Equivalence

In a population of 10 people, let \(R \) be the "older than" relation and \(T \) be the "taller than" relation.

1. Which of the following properties guarantee that \(R \) will be a linear order?

   Exercise 1

     There is a unique oldest person There are at most two people with the same age No two people are the same age There is an age that no one has Some person appears twice in the list No two people are the same age 



2. Assume both \(R \) and \(T \) are linear orders. Which properties are guaranteed to be true for the product relation \(R \times T \)?

   Exercise 2

     symmetric  

     antisymmetric  

     asymmetric  

     reflexive  

     transitive  

     acyclic  

     linear  

   Not symmetric because two distinct people cannot both be older and taller than each other.

   Not reflexive because a person cannot be older/taller than himself.

   Not linear because if person \(A \) is older than person \(B \) but \(B \) is taller than \(A \), then \(A\) and \(B\) are incomparable.

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