Mutually exclusive events, pairwise mutually exclusive events and disjoint sets I have been confused by the phrases: mutually exclusive events, pairwise mutually exclusive events and disjoint sets. My book uses these at different places. What is the difference between them?
 A: The concepts are closely related but simple.  This suggests that a small example might help clarify them.
The Example
When we talk about "events" we are working in a context where we intend to associate probabilities to some--but perhaps not all--sets of outcomes.  Thus, the stage is set by specifying two things:

*

*$\Omega$ is a set of outcomes, the "sample space" of our study.


*$\mathfrak F$ is a collection of subsets of $\Omega$ to which we might want to assign probabilities.
A nice example is afforded by the binomial asset price model of mathematical finance, as described in other threads such as https://stats.stackexchange.com/a/123754/919.  Briefly, suppose you are interested in tracking the price of an asset over time.  A very simple model supposes it might go up ($+$) or down ($-$) tomorrow and, separately, it can also go up or down the day after.  We might indicate the resulting four possibilities by the set $\Omega = \{++, +-, -+, --\}$ where the first character in each string indicates what happens tomorrow and the second character indicates what happens the day after tomorrow.
Investors make decisions over time.  In this model we can represent an investor's knowledge tomorrow by grouping all outcomes corresponding to a common state of knowledge.  That is, if tomorrow the price goes up, there are two possible ultimate outcomes, $++$ and $+-;$ and otherwise, the two possible outcomes are $-+$ and $--.$  Let's call these sets $H_+ = \{++, +-\}$ and $H_- = \{-+, --\},$ respectively.  For the purposes of modeling what can be known tomorrow, the investor will associate probabilities with $H_+$ and $H_-,$ as well as with all logical combinations of them achievable by taking unions, intersections, and complements.  This implies probabilities can be associated with any member of the set algebra
$$\mathfrak F = \{\emptyset, \Omega, H_+, H_-\},$$
but the remaining 12 subsets of $\Omega$ will just never have probabilities at all, because in the application they are not relevant to tomorrow's information.
Answers
With this example in mind, here are the answers to your questions:

*

*Events are the particular sets to which probabilities can be assigned (called "measurable sets").  In the example, there are 16 subsets of $\Omega$ but only 4 of them are events.

*

*An example is the set $H_-.$


*A non-example is the set $\{++, +-, -+\}.$  This is not in $\mathfrak F.$  It can have no probability assigned to it.




*Disjoint sets do not intersect.  Specifically, any sets $A$ and $B$ are disjoint when $A\cap B = \emptyset.$

*

*An example is $\{++, +-, -+\}$ and $\{--\}:$ these sets have no element in common.


*A non-example is $\{++, +-, -+\}$ and $H_-$: they both contain $-+.$




*Pairwise disjoint sets are disjoint in pairs.  Consider any collection of sets $A_i$ where $i$ ranges over some index set $\mathcal I.$  When for every $i,j\in\mathcal I$ with $i\ne j,$ $A_i$ and $A_j$ are disjoint, we say the collection $(A_i)_{i\in\mathcal I}$ is pairwise disjoint.

*

*An example is the collection $\{\emptyset, H_-, H_+\}.$  Any two of these three sets have no elements in common.


*A non-example is $\mathfrak F$ itself (consisting of four sets).  Although the intersection of all the sets in $\mathfrak F$ is empty (no outcome is contained in all events), some pairs of events in $\mathfrak F$ have nonempty intersection (such as $\Omega$ and $H_+,$ for instance).




*Mutually exclusive events are events that, as mere sets, are disjoint.


*Pairwise mutually exclusive events are collections of events that, as mere collections of sets, are pairwise disjoint.
For more discussion of the set-related concepts see the Wikipedia article on disjoint sets.
