I am reading Lebesgue measure. In many situations I have found that the author says pairwise disjoint collection of subsets of $\mathbb{R}$ and in some others simply disjoint. What is the difference in Mathematics?
Asked
Active
Viewed 1.5k times
3 Answers
9
A set (of sets) $\mathcal{A}$ is disjoint if $\bigcap \mathcal{A} = \emptyset$.
The set $\mathcal{A}$ is pairwise disjoint when $\forall x \in A: \forall y \in A: x \neq y \implies x \cap y = \emptyset$. This implies disjoint if $|\mathcal{A}| \ge 2$.
So $\mathcal{A} = \{x,y\}$ is disjoint iff it is pairwise disjoint.
But in measure theory, disjoint is often used as a shorthand for "pairwise disjoint".
Henno Brandsma
- 242,131
-
2Do you have any reference to back up your first sentence? This book says that "disjoint" means pairwise disjoint. – Marc van Leeuwen Mar 09 '19 at 12:36
-
@MarcvanLeeuwen it’s often used as a short term for pairwise disjoint. It depends on the text. I cannot check my books for a reference now. – Henno Brandsma Mar 09 '19 at 12:51
-
@MarcvanLeeuwen the historical use of the word "disjoint" was "empty intersection". A philological argument is that this is why the expression "pairwise disjoint" even exists! There would be no need to use the word "pairwise" if there were not other ways to be disjoint. – Rad80 Mar 11 '19 at 10:56
-
1@Rad80 What makes you give your historical claim? Of course for two sets, disjoint means empty intersection, but I have yet to see a single mathematical text (not on an internet forum like this one) where disjoint for more than two sets is taken to mean empty intersection. Interestingly at this answer somebody did find an early historic use which is not easy to read, but where I found disjoint for collections to mean "pairwise distinct" (see my comments there). As for the philological argument , it proves nothing; the "pairwise" is just emphasis. – Marc van Leeuwen Mar 11 '19 at 11:24
7
Usually there is no difference in meaning. Sets $A_1$, ..., $A_n$ are (pairwise) disjoint if $A_i \cap A_j = \emptyset$ whenever $i \neq j$.
user133281
- 16,073
0
The term disjoint refers to a collection of subsets, it means that its subsets are disjoint.
The term pairwise disjoint refers to a familly of collections of subsets. It not only means that any two collections of that family are disjoint, i.e. they share no common element (that is, a common subset), but that in addition, if you take one set from each of the two collections, then these sets will be not only distinct, but also disjoint.
