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