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I've just started trying to learn some set theory and topology and I've come across the definition of disjoint sets quite a lot I've seen lots of Definitions such as

A set (of sets) $\mathcal{A}$ is disjoint if $\bigcap \mathcal{A} = \emptyset$.

The set $\mathcal{A}$ is pairwise disjoint when $\forall x \in A: \forall y \in A: x \neq y \implies x \cap y = \varnothing$

I can't quite understand what the difference actually is as I saw on s.e that a pairwise disjoint set is related to a $k$-wise disjoint.

I also saw this definition

An indexed collection $\{A_i\}_{i\in I}$ of subsets of $X$ is said to be pairwise disjoint if $A_i\cap A_j=\varnothing$ whenever $i\neq j$.

This confused me even further as an indexed collection can be a surjective function so when $i=j$ it could well be that $A_i$ = $A_j$ in that case how is their intersection empty?.

What I'm really trying to understand is what is pairwise and k wise disjoint sets? Thanks in advance

Asaf Karagila
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The homeschooler
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    ${A,B},{B,C},{C,A}$ (where $A\neq B\neq C$) are disjoint, since there is no element common to all three sets, but are not pairwise disjoint, since each two of them have a common element – Alessandro Codenotti Feb 04 '21 at 20:41
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    In the indexed collection version there is often (as here) that the indexing is faithful which means that distinct indices imply different sets. – Henno Brandsma Feb 04 '21 at 20:52
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    The notion of three or more sets being disjoint is not very useful. Disjointness is really a property of two sets; and although there is only one reasonable way to define disjointness of three or more sets, it is more natural simply to say that the sets have no common element, or that their intersection is empty. – TonyK Feb 04 '21 at 21:14
  • @henno brandsma is that to say that faithful means the function is injective – The homeschooler Feb 04 '21 at 23:34

2 Answers2

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To supplement the other answers and comments, let me summarize my view.

An indexed family $(A_i)_{i ∈ I}$ of sets is

  • pairwise disjoint if $A_i ∩ A_j = ∅$ for every $i ≠ j ∈ I$,
  • collectionwise disjoint if the family has empty intersection, i.e. $⋂_{i ∈ I} A_i = ∅$.

The term disjoint may serve as a shortcut either for pairwise disjoint or collectionwise disjoint depending on the used convention (but obviously not both at the same time). It seems to me that “classically” it is used for collectionwise disjoint, but at least to me (and @TonyK) it seems more useful to use it for pairwise disjoint and to call a collectionwise family just a family with empty intersection.

Note that if an indexed family of non-empty sets is pairwise disjoint, then the map $i ↦ A_i$ is one-to-one, i.e. the enumeration is faithful (as @Henno Brandsma says). But a non-empty indexed family of empty sets $(∅)_{i ∈ I ≠ ∅}$ is pairwise disjoint even though the enumeration is not faithful.

Also note that pairwise disjointness implies collectionwise disjointness unless $I = ∅$.

An unindexed family of sets $\mathcal{A}$ is whatherverwise disjoint if the indexed family $(A)_{A ∈ \mathcal{A}}$ is.

user87690
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  • thank you very much this has cleared my confusion – The homeschooler Feb 04 '21 at 23:36
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    Great summary!$\quad$ To add: 'mutually disjoint' and 'pairwise disjoint' are synonyms (Brian M Scott agrees); after all, both 'mutual independence' and 'mutual respect' too imply pairwise. $\quad$ I have a parallel discussion of mutual exclusivity here. $\quad$ And here, I point out that, due to the ambiguous phrase "if any", wikipedia's definition of 'disjoint set' simultaneously accomodates both the pariwise & collectionwise interpretations! – ryang Feb 28 '23 at 18:59
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$A$ is the set of even numbers, $B$ is the set of odd numbers, and $C$ is the set of multiples of $3$. The 3 sets are collectively disjoint (i.e. no element is in all 3 sets), but not pairwise disjoint, because (for example) the number $3$ is in both sets $B$ and $C$.

user2661923
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