left and right euclidean;
A (binary) relation on a set is irreflexive if no element of is related to itself:
\forall (x: A),\; x \nsim x .
\id_A \cap R \subseteq \empty .
Of course, this containment is in fact an equality.
\forall (x: A),\; \forall (y: A),\; x \sim y \;\Rightarrow x \# y .
Since is irrelexive itself, any strongly irrelexive relation must be irrelexive. The converse holds using excluded middle, through which every set has a unique tight apartness.