5. a) For all ordered pairs (a, b) in Z × (Z \ {0}), (a, b) r (a, b) since a/b = a/b , so the relation r is reflexive.
b) For all ordered pairs (a, b) and (c, d) in Z × (Z \ {0}), if (a, b) r (c, d), then a/b = c/d , which implies that c/d = a/b, so (c, d) r (a, b). Thus, the relation r is symmetric. (It might help to start thinking about this as a particular example, such as 1/2 = 3/6, so 3/6 = 1/2, and then thinking about it in general terms. )
c) For all ordered pairs (a, b), (c, d) and (e, f) in Z × (Z \ {0}), if (a, b) r (c, d) and (c, d) r (e, f), then a/b = c/d and c/d = e/f , which implies that a/b = e/f, so (a, b) r (e, f). Thus, the relation r is transitive. (It might help to start thinking about this as a particular example, such as 1/2 = 3/6 and 3/6 = 5/10 so 1/2 = 5/10 , and then thinking about it in general terms. )