Pairwise disjoint or disjoint?

Question

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?

Answer 1

Usually there is no difference in meaning. Sets $A_1$, ..., $A_n$ are (pairwise) disjoint if $A_i \cap A_j = \emptyset$ whenever $i \neq j$.

Answer 2

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.

Answer 3

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".