Let A, B and C be any three events. Show that

(i) if and only if

(ii) Given and , determine

Show that for any two events :

(i) If , then

(ii) If , then

(iii) Why is it incorrect to assume that for some events A and B, and ?

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- Mar 14th 2009, 08:45 PMpanda*Axioms Proving.
Let A, B and C be any three events. Show that

(i) if and only if

(ii) Given and , determine

Show that for any two events :

(i) If , then

(ii) If , then

(iii) Why is it incorrect to assume that for some events A and B, and ? - Mar 15th 2009, 01:41 AMMoo
Hello,

I'll give you most of the solution, but you will have to fill in some steps ;)

Thing that may come in handy :

(de Morgan's law)

(de Morgan's law)

From here, it should be very easy to conclude :)

Quote:

(ii) Given and , determine

Quote:

Show that for any two events :

(i) If , then

why ? because let's consider an element in B. It is contained in A,**or**it is contained in B, but not in A. This latter possibility gives the set

You can also see that since A and are disjoint, then A and are disjoint.

Hence

And the conclusion follows.

Quote:

(ii) If , then

but since is a probability, it's

thus

Quote:

(iii) Why is it incorrect to assume that for some events A and B, and ?

By (ii), we should have , which is not the case here ;)