# 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?

## 2 Answers

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).

• "In math, either a statement is true, or its negation is true (but not both)" actually, this is only valid in classical logic, but it really doesn't matter for this question Commented Apr 7, 2020 at 19:19

If $A$ is not a family of sets which are pairwise disjoint, then there exists $A_1,A_2\in A$ such that $A_1\neq A_2$ and $A_1\cap A_2\neq\varnothing$.

So... yes.