GRE 9367 #60

[aas56]

Let A and B be subsets of M and let S(0)={A,B}. for i>= 0 define S(i+1) inductively to be the collection of subsets X of M that are of the form CUD , C inter D , M-C, where C, D are in S(i). let S be the union of all the S(i). What is the largest number of elements of S(i)?
A)4 B)8 C) 15 D)16 E)infinity


I tryed to solve this question by counting the sets and I always get 14 elements :
A,
B,
empty set,
M-A,
M-B,
M,
AUB,
A inter B,
AU(M-B),
Aint(M-B),
(M-A)UB,
(M-A)interB,
(M-A)U(M-B),
(M-A)inter(M-B)
The right answer happened to be D. What are the remaining uncounted set?

[weilan8]

I think there are two sets that are omitted:
(A-B)U(B-A)
(complement(A)-complement(B))U(complement(B)-complement(A))

[mag487]

Draw a Venn diagram of A, B and M. You'll notice there are four disjoint regions which cover the entire picture: the things in M outside of both A and B, the things in A alone, the things in B alone and the things in both A and B. Every element of the algebra over {A, B} either includes or excludes each of these four regions. Thus there are 2^4 = 16 elements of the algebra.

[breezeintopl]

Draw a Venn diagram of A, B and M. You'll notice there are four disjoint regions which cover the entire picture: the things in M outside of both A and B, the things in A alone, the things in B alone and the things in both A and B. Every element of the algebra over {A, B} either includes or excludes each of these four regions. Thus there are 2^4 = 16 elements of the algebra.

cool!