Expert: Sherry Wallin - 3/19/2009

Question
Hi,
I have to write an application problem using a Venn diagram
with three different sets and describe what
n(A'intersectedBintersectedC) results in for the
application.

Secondly, using the Cartesian product and the
multiplication principle, I have to describe what AxBxC,
the cartesian product for three sets A, B, C would look
like. How do I give an example of finding the product to
demonstrate the idea? How do I give a formula for counting
the number of elements in the set AxBxC? How could I prove
it's correct?

I've been stressing over these problems for a few days now.
I've tried a few different ways but everything comes out
wrong. Thank you for your time,
Lu

Answer
Hi Lu~
    It is not clear to me what your first question is. Are you asking me to describe A intersect B intersect C?

For the 2nd part, the Cartesian product of three sets is thus:
Suppose A ={1,2} and B = {2,4} and C = {1,3}, then AxBxC =
(1,2,1), (1,2,3), (1,4,1), (1,4,3),(2,2,1), (2,2,3), (2,4,1), (2,4,3)
What you are doing is looking at all the different combinations of members of set A in the first position of the ordered triplet, members of set B in the 2nd position of the ordered triplet, and members of set C in the 3rd position of the ordered triplet. Just like the ordered pairs (x,y) these are like (x,y,z) where x belongs to A, y belongs to B, and z belongs to C. How do you determine how many? The are 2 choices from A and 2 choices from B and 2 choices from C so that is 2(2)(2)= 2^3 ordered triplets in the Cartesian product. If each set A,B,C had 3 elements there would have been 3 choices from each or 3^3 = 27 ordered triplets, so if the sets A,B,C had n elements there would be n^3 ordered triplets. To prove this, use induction. Show it is true (that there is 1 triplet if each of A,B,C has 1 element) for n = 1, then assume it is true for n <= k and then show through algebraic manipulation that it is true for n = k + 1.  

As for Venn Diagrams, if an element belongs to all three sets then it will be in the intersection (the center where all three sets overlap).

Hope this helped you,

Math Prof