Are pairwise mutually exclusive events the same as mutually exclusive events?
Larson (1982) defining the probability axioms talks about "mutually exclusive" events, while Poirier (1995) about "$A_1, A_2, \ldots$ as a sequence of pairwise mutually exclusive events events in the sigma-algebra $\tilde A$."
I suppose that the two notions are equivalent (they both refer two disjoint sets), right? Does this make adding the word "pairwise" superfluous on behalf of Poirier?
Is there any other context out of probability that makes this distinction (using the word pair-wise) meaningful? According to wikipedia, in Logic, "pairwise mutually exclusive" means that both propositions cannot be true simultaneously, in contrast to just mutually exclusivity that means that if one is true, then the other cannot be true.
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$\begingroup$What about vicious cycles? For example, suppose you have three binary random variables $A$, $B$, and $C$ that each take values $0$ or $1$. In this example, $A$ and $B$ are mutually exclusive if whenever $A = 0$, $B = 1$, and similarly for $C$.
$A$, $B$, and $C$ cannot all be pairwise mutually exclusive.
However, they can be "globally" mutually exclusive in the sense that, for example, if $A = 0$, then $B = 1$ and $C = 1$, and so on for the other variables.
I think that this counterexample proves that pairwise mutual exclusivity is not equivalent to mutual exclusivity.
$\endgroup$ $\begingroup$They don't mean the same thing, but it is a short jump to show they are equivalent. A set of events, $A_1,...,A_n$ are mutually exclusive if the occurrence of one of them implies that the other $n-1$ events can't happen. It's immediate that the events are pairwise mutually exclusive. The other direction is immediate as well.
They aren't a priori the same, but it's immediate that they're equivalent.
$\endgroup$ $\begingroup$Sets $A_1, A_2, ...$ are mutually disjoint if
$$\bigcap_{i=1}^{\infty} A_i = \emptyset $$
Sets $A_1, A_2, ...$ are pairwise disjoint if
$$A_i \cap A_j = \emptyset \ \forall i \ne j$$
Apparently, most texts use 'disjoint' to refer to 'pairwise disjoint'. Whenever a text uses 'pairwise disjoint', we can assume 'disjoint' refers to 'mutually disjoint'
In our case, 'mutually exclusive' by Larson means the same thing as 'pairwise mutually exclusive'. It depends on the text I guess.
See my questions:
Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?
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