Events, Sample spaces, Sets and boolean logic

mikau · 2008-09-09 09:20:12

So my discrete math book, in the section on sets suddenly begins using the terms 'event' and 'sample space' without defining them.

I think I can guess the idea. It sounds like an 'event' is a set of possible occurrences, and an occurrence is thus an instance of an event. Or an element of an event.

What a sample space is i'm not sure. I seem to recall that term from probability.

Anyway, so the impression I get is that we have a kind of event, such as the event of a building fire, and the event of building collapse. And we can therefore consider these as sets and take the intersection, to 'AND' both events together, to get the event that a building is on fire and collapses.
Moreover, we can take the union to 'OR' both events together, to get the event that either the building is on fire or collapses. Lastly, we can negate or 'NOT' an event by using the 'without' set operation, for instance, the event in which the building is on fire, without the collapsing event

so it looks like you can construct logical statements about events using intersection, union, and 'without' (or complement) for AND, OR, and NOT.

So here's two problems, please let me know if my approach is correct:

Let A B C be events of some sample space Q. Write in symbols:
(1) The event that at least two of the three events occurs
(2) The event that at most one of the three events occurs

(1), we have that either 2 of the events occur, or all 3,
two events means (A and B) or (A and C) or (B and C) or (A and B and C)
the last is redundant though it shouldn't hurt anything,
so for set operations we replace 'and' with 'intersection' and 'or' with 'union' and we should get a valid answer, right?

(2) only one most occur at a time, so we have A and not (B or C) or B and not (A or C) or C and not (A or B)

we can replace 'and' and 'or' with 'intersects' and 'union' and not (in this case) with 'without'.

Is this the correct approach to these problems?

If so, this relation between set theory and boolean algebra is pretty sweet.

Last edited by mikau (2008-09-09 09:22:50)

John E. Franklin · 2008-09-09 19:08:34

Looks right to me Mikau! Your #(2) can also be expressed
using XOR logic operators additonally like this:

( (A XOR negC) and (negC or negB) ) XOR negB

Its #172 grid at the webpage, or #11, depending how you manipulate it.

Math Is Fun Forum

#1 2008-09-09 09:20:12

Events, Sample spaces, Sets and boolean logic

#2 2008-09-09 19:08:34

Re: Events, Sample spaces, Sets and boolean logic

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