Must antisymmetric relation also be irreflexive

Question

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?

Answer 1

No, an antisymmetric relation doesn't have to be irreflexive. However, if a relation $R$ is both transitive and irreflexive, then it is automatically antisymmetric. Here is the reason:

Say we have $(a,b)\in R$. Then, because of transitivity, if we also have $(b,a)\in R$, we must have $(a,a)\in R$, which breaks irreflexivity. Therefore we can never have both $(a,b)$ and $(b,a)$ in $R$, which means that the statement "If $(a,b)\in R$ and $(b,a)\in R$, then $a=b$" is vacuously satisfied, and $R$ is thus antisymmetric.

Answer 2

It's a common misunderstanding of “if”. Consider the following true statement about Alice, who's outside at the beach:

Alice gets wet if it rains

Does it need to rain in order that Alice gets wet? Of course not! She could be swimming in the sea in a very hot sunny day!

Now change “Alice gets wet” with “the relation $R$ is antisymmetric” and “it rains” with “the relation $R$ is irreflexive and transitive”: you get the statement on Wikipedia.

The statement says, expressed in a different way: a sufficient condition for a relation to be antisymmetric is to be irreflexive and transitive.

The statement doesn't say that this condition is also necessary. Indeed, a trivial example is the relation $\{(1,1),(2,2)\}$ on the set $\{1,2\}$ (the equality).

More generally, a partial (non strict) order is a reflexive, antisymmetric and transitive relation. We surely don't want it to be irreflexive, do we? ;-)

There is a relation that's reflexive and irreflexive, but it's a very boring one: the empty relation on the empty set.