1.12 There is a slight distinction between the notions of pairwise disjoint for nonindexed collections of sets and indexed collections of sets, namely, an indexed collections of set $\{A_i\}_{i\in I}$ can fail to be pairwise disjoint even though the collection, $C=\{A_i:i\in I\}$, is pairwise disjoint. Provide an example that illustrates this fact.
Two subsets $A$ and $B$ of $X$ are said to be disjoint if $A\cap B=\varnothing$.
A collection $C$ of subsets of $X$ is said to be pairwise disjoint if each two distinct members of $C$ are disjoint.
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 is a question from a book A course in real analysis, however I cannot find any example to verify the the distinction between different notions, I always think they are the same, even though some very special case such as index set is empty.
Best Answer
If two sets $A,B$ are taken from collection $\mathcal C$ and $A\neq B$ implies in that situation that $A\cap B=\emptyset$ then the (non-indexed) collection $\mathcal C$ can be labeled as 'pairwise disjoint'. Looked at as an indexed collection more is needed: $i\neq j$ must imply that $A_i\cap A_j=\emptyset$
Underlying is the fact that you can have two distinct indices $i\neq j$ with $A_i=A_j$.
Example:
Take $A_i$ for $i=1,2,3$ and $A_1=A_2\neq\emptyset$ and $A_1\cap A_3=\emptyset$.
Then collection $\{A_1,A_2,A_3\}=\{A_1,A_3\}$ is pairwise disjoint, but indexed collection $\{A_i\}_{i\in\{1,2,3\}}$ is not.