On Polish Wikipedia article on binary relations one can find the following statement: "a relation is antisymmetric iif it is irreflexive and transitive".
Is it correct? Does a given relation have to be irreflexive to be antisymmetric?
As far as I understand a relation ($R \subseteq A \times A$) is irreflexive iif: $$\forall_{x \in A}: \lnot (x R x) $$ and antisymmetric iff: $$\forall_{x, y \in A}: (xRy) \land (yRx) \rightarrow x = y$$ As so, if we define relations over the set $A = \{1, 2, 3, 4\}$ then both following relations should be antisymmetric: $$ R_1 = \{ (2,1), (3,1), (3,2), (4,1), (4,2), (4,3) \}\\ R_2 = \{ (1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4) \} $$ although only $R_1$ is also irreflexive ($R_2$ is actually reflexive).
Do I miss something?