If we roll a die, the obvious sample space is. If we can observe and are interested in the exact outcome, then we may consider singleton events consisting of a single outcome,

The natural probability measure assigns 1/6 with each outcome, but we would use a different measure in the case of an unfair die. However, we might also be interested in assigning a probability to other events as well. The probability that we observe 2 or 3 should be associated with the event , and intuition suggests that

In this case, since the two events are mutually exclusive, we just add probabilities. More generally, given events and, we might be interested in the probabilities

where we see a natural connection with set operations like union, intersection, and difference. Note that or is related to an inclusive "or," rather than to the exclusive "either... or..." (but not both). By applying arbitrary combinations of these operations to singletons and composite events, we may generate a rather large family of all subsets of cardinality 1,2 , etc., also including itself and its complement, the empty set :

The familyof all subsets of  Ω is clearly closed with respect to the set operations, as by applying set operations to subsets in, we can only generate an element in .

However, we may constrain events a bit in order to reflect the possibly limited amount of information. For instance, we might consider the following family of events:

which makes sense when all we may observe (or are interested in) is whether the result is odd or even. This family of events, with respect to, is definitely less rich, and this reflects lack of information. However, it is easy to check that if we try taking complements and unions of elements in, we still get an element of .

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= {1, 2, 3, 4, 5, 6}