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
Best Answer
To supplement the other answers and comments, let me summarize my view.
An indexed family $(A_i)_{i ∈ I}$ of sets is
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.