chap mathematical preliminaries also discusses the three proof techniques presented in section and the roles of investigation and argument in problem solving for more about estimating techniquessee two programming pearls by john louis bentley entitled the back of the envelope and the envelope is back [ben ben ben ben geniusthe life and science of richard feynman by james gleick [gle gives insight into how important back of the envelope calculation was to the developers of the atomic bomband to modern theoretical physics in general exercises for each relation belowexplain why the relation does or does not satisfy each of the properties reflexivesymmetricantisymmetricand transitive ( "isbrotherofon the set of people ( "isfatherofon the set of people (cthe relation {hxyi for real numbers and (dthe relation {hxyi for real numbers and (ethe relation {hxyi mod for xy { (fthe empty relation ( the relation with no ordered pairs for which it is trueon the set of integers (gthe empty relation ( the relation with no ordered pairs for which it is trueon the empty set for each of the following relationseither prove that it is an equivalence relation or prove that it is not an equivalence relation (afor integers and ba if and only if is even (bfor integers and ba if and only if is odd (cfor nonzero rational numbers and ba if and only if (dfor nonzero rational numbers and ba if and only if / is an integer (efor rational numbers and ba if and only if is an integer (ffor rational numbers and ba if and only if | < state whether each of the following relations is partial orderingand explain why or why not ( "isfatherofon the set of people ( "isancestorofon the set of people ( "isolderthanon the set of people ( "issisterofon the set of people ( {habihaaihbaion the set {ab