Sara has devised a system for rating the males in her class. She cares about their intelligence and their looks. She has ranked each male on a scale of 1 to 5 for intelligence (5 meaning "brainiac") and 1 to 3 for looks(3 meaning "dreamy"). She defines a preference relation,
R, as follows: xRy if boy x scores at least as high as boy y in either looks or intelligence. Provide an intuitive argument (if the answer is "yes") or counterexample (if the the answer is "no") to each of the following:
(a) Is R complete?
(b) Is R transitive?
(c) Is R rational?
I know it is rational if it is both complete and transitive. I think it is complete, but I'm not sure if it is transitive and I'm also not sure what an intuitive argument means exactly. Thanks!