It is written as A1 ∩ A2 ∩ A3 = ∅, and sorry for all the questions. First week of class and my Professor hasn't lectured on anything that helps with these Problems, so I have been trying to get help here, and read a lot on the internet. He also has no text for the course so I have been trying to find a good text for Discrete Math. Any suggestions would be appreciated.
Well I really feel for you in
case.
If it were the case that you have three nonempty pairwise disjoint sets the answer is 9330. That is known as a
Stirling number of the second kind: \(\displaystyle S_2(10,3)=9330\). The number of ways to partition a set of ten into three cells.
Of course that means \(\displaystyle \left\{ {\{ 0,1,2,3\} ,\{ 4,5,6\} ,\{ 7,8,9\} } \right\} = \left\{ {\{ 7,8,9\} ,\{ 0,1,2,3\} ,\{ 4,5,6\} } \right\}\).
That is there is no order to the cells of a partition.
NOW if he does mean ordered triples the number is multiplied by \(\displaystyle 3!=6\).
On the other hand, if we could have \(\displaystyle \left\{ {A_1 = \{ 0,1,2,3\} ,A_2 = \{ 3,4,5,6\} ,A_3 = \{ 1,2,6,7,8,9\} } \right\}\)
We do have \(\displaystyle A_1\cap A_2\cap A_3=\emptyset. \).
And I don't know right off how one counts those triples.
As to the matter of a textbook. In the last twenty-five years there has been an explosion in good discrete mathematics textbooks. But each has a slightly difference focus. I like James Anderson's text for a counting focus. The best balanced text is by Ross & Wright.
