Answer to Question #282704 in Discrete Mathematics for bennie

Question

Show that the relation R = ∅ on a nonempty set S is symmetric and transitive but not reflexive.

Answer 1

A relation R over a set S is reflexive if and only if for any element x of S,  xRx.

So you have a nonempty set S, so take an element x. As R is empty, there is no y such that xRy. In particular it’s not true that xRx, so R is not reflexive.

On the other side, both symmetry and transitivity are defined over the relation itself:

• symmetry: xRy implies yRx
• transitivity: xRy and yRz imply xRz

so these implications are always true in an empty relation, because the premise never holds.