 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Set Me Up (Posted on 2004-02-10) 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 = {} ?

 See The Solution Submitted by DJ Rating: 4.3636 (11 votes) Comments: ( Back to comment list | You must be logged in to post comments.) re(4): Solution FINAL WORD | Comment 27 of 31 | My formula is f(n) = 6^n - (3*3^n - 3) if no empty sets are allowed.
f(1) = 0 is correct since at least one of A, B and C must be empty for the intersection to be empty.

f(2) = 36 - (3*9 - 3) = 12. The triples (A,B,C) are:
({1},{1},{2})
({1},{2},{1})
({2},{1},{1})
({1},{2},{2})
({2},{1},{2})
({2},{2},{1})
({1,2},{1},{2})
({1,2},{2},{1})
({1},{1,2},{2})
({2},{1,2},{1})
({1},{2},{1,2})
({2},{1},{1,2})

f(3) = 138 and f(4) = 1056 are longer to list.
f(3) = 216 - (3*27 - 3) = 138
f(4) = 1296 - (3*81 - 3) = 1056
 Posted by Brian Smith on 2004-02-10 15:13:44 Please log in:

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