Why do we say ‘pairwise disjoint’, rather than ‘disjoint’?

I don’t see the ambiguity that ‘pairwise’ resolves.

Surely if $A$, $B$ and $C$ are disjoint sets then they are pairwise disjoint and vice versa?

Or am I being dim?

Solution 1:

$\{1,2\},\{2,3\},\{1,3\}$ are disjoint but not pairwise disjoint.

Solution 2:

As evidenced by the answers and comments on this page, the term "disjoint" is ambiguous - some use it to mean "pairwise disjoint", others use it to mean "empty intersection".

Thus, for the sake of clarity, I'd recommend avoiding "disjoint" and using "pairwise disjoint".

Solution 3:

The word "pairwise" in "pairwise disjoint" is superfluous: a collection of sets is disjoint if no element appears in more than one of the sets at a time, and this means that every pair of distinct sets in the collection has an empty intersection. However including the "pairwise" emphasizes that the property can be checked at the level of pairs from the collection (unlike for instance linear independence of vectors in linear algebra). A "disjoint union" is a union of pairwise disjoint sets; one does not say "pairwise disjoint union".

To corroborate my point of view, here is a citation from Halmos:

Pairs of sets with an empty intersection occur frequently enough to justify the use of a special word: if $A\cap B=\emptyset$, the sets $A$ and $B$ are called disjoint. The same word is sometimes applied to a collection of sets to indicate that any two distinct sets of the collection are disjoint; alternatively we may speak in such a situation of a pairwise disjoint collection.

By the way, it is amazing how this site contains lots of answers saying disjoint (for any collection of sets) means empty intersection, like at this question and questions linked from there, and lots of comments saying that is wrong. Which was my motivation to post this as an answer.