Q:

Is the empty family of sets pairwise disjoint?

„A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.“ – from Wikipedia article "Disjoint sets"
What about the empty family of sets? Is it also pairwise disjoint?
I think, that the empty family of sets is pairwise disjoint, because statements of the form $\forall x \in \emptyset:\ldots$ are always true. Am I right?

A:

Yes, you are right.  It is vacuously true.  Here's a more detailed explanation of why:
In math, either a statement is true, or its negation is true (but not both).  That means, for example, either the statement (a) $\forall x \in \emptyset$, $x^{2} = 1$ or its negation, (b) $\exists x \in \emptyset$ such that $x^{2} \neq 1$, is true, and the other is false.
It's clear that statement (b) is false since $\exists x \in \emptyset$ is a false statement.  So, since statement (b) is false, its negation, statement (a), must be true (it's called vacuously true).

