To demonstrate set union and intersection to her class, Mrs. Putnam asked for three students to each write down a set of numbers.

After they had done so, she looked at their sets and told the class, "the union of these three sets is the first ten counting numbers, but their intersection is empty!"

How many triples (A, B, C) of sets are there such that

A U B U C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and
A ∩ B ∩ C = {} ?

(In reply to Is this right ? by Penny)

Penny: "If C has 2 numbers (there are 10*9 such sets), then any two sets A and B, where A is any subset of the remaining 8 numbers, and B is any subset of all 10 numbers, should do the trick."

Penny doesn't make sense. These combinations include
C={1,2}, B={1,2,3,4}, A={3,4}.
A U B U C = {1,2,3,4}

It is also wrong to assume that if there are three subsets A,B,C such that A U B U C = {1,2,3,4,5,6,7,8,9,10} and
A int B int C = {}, then at least two of them must have null intersection.

What about:
A = {1,3}, B={1,2}, C={2,3,4,5,6,7,8,9,10}

At this rate I'll have the cure for cancer any day now....
Edited on February 10, 2004, 4:49 pm

Posted by Penny on 2004-02-10 16:47:23