Final answer:
The relations 'x - y is a rational number' and 'xy = 1' both exhibit symmetry and transitivity. The relation 'a is a multiple of y' is antisymmetric and transitive. The reflexive property does not hold consistently across these relations.
Step-by-step explanation:
Let's analyze the given relations on the set of all real numbers.
(b) x - y is a rational number
This relation is reflexive, because for any real number x, x - x = 0, which is rational. It is symmetric because if x - y is rational, then y - x is also rational (since the negative of a rational number is rational). However, it is not antisymmetric because there could exist distinct x and y such that both x - y and y - x are rational.
The relation is transitive; if x - y and y - z are rational, then (x - y) + (y - z) = x - z is also rational.
(f) xy = 1
This relation is neither reflexive (0*0 does not equal 1) nor antisymmetric (if xy = 1, then yx = 1, which means x and y could be distinct and still related). It is symmetric since if xy = 1, then yx = 1. It is transitive; if xy = 1 and yz = 1, then xz = 1.
(g) a is a multiple of y
The relation 'a is a multiple of y' is not reflexive (0 is not a multiple of 0), not symmetric (if a is a multiple of b, does not imply b is a multiple of a), but is antisymmetric (if a is a multiple of b and b is a multiple of a, then a = b, except for the case of 0, where we can define it as needed).
It is transitive; if a is a multiple of b and b is a multiple of c, then a is a multiple of c.