„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?


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).


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.


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