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„A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.“ – fromWikipedia 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?

askedMar 29, 2015 at 15:37
Stephan Kulla's user avatar
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2 Answers2

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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).

answeredMar 29, 2015 at 15:42
layman's user avatar
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  • $\begingroup$"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$\endgroup$CommentedApr 7, 2020 at 19:19
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If $A$ isnot a family of sets which are pairwise disjoint, then thereexists $A_1,A_2\in A$ such that $A_1\neq A_2$ and $A_1\cap A_2\neq\varnothing$.

So... yes.

answeredMar 29, 2015 at 15:42
Asaf Karagila's user avatar
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