Question: The terms "reflexive", "symmetric" and "transitive" were defined in Footnote 2. Which of these properties is satisfied by the relationship of "greater than?" Which of these properties is satisfied by the relationship of "is a brother of?" Which of these properties is satisfied by "is a sibling of?" (You are not considered to be your own brother or your own sibling). How about the relationship "is either a sibling of or is?"
a) Explain why an equivalence relation (as we have defined it) is a reflexive, symmetric, and transitive relationship.
b) Suppose we have a reflexive, symmetric, and transitive relationship defined on a set S. For each x is S, let Sx = {y|y is related to x}. Show that two such sets Sx and Sy are either disjoint or identical. Explain why this means that our relationship is an equivalence relation (as defined in this section of the notes, not as defined in the footnote).
c) Parts b and c of this problem prove that a relationship is an equivalence relation if and only if it is symmetric, reflexive, and transitive. Explain why. (A short answer is most appropriate here.)