1. Subsets and Combinatorics

What is the number of subsets (including the empty set) of {1,2,3,...,12} that are contained in (at least) one of the following sets:

{1,2,3,4,5,6}, {5,6,7,8}, {7,8,9,10,11,12}, and {1,6,11,12}

Any ideas would be greatly appreciated. Thanks for your time.

2. You could try using the Inclusion–exclusion principle and note that $\mathcal{P}(A)\cap\mathcal{P}(B)=\mathcal{P}(A\cap B)$.

3. Using the Inclusion-Exclusion Principle:

2^6 + 2^4 + 2^6 + 2^4 - 2^2 -2^2 - 2^2 - 2^2 - 2^1 - 2^0 + 2^0 + 2^0 + 2^1 - 2^0 = 144

However, I also wrote out all the possible subsets, counted and removed the duplicates and found the answer to be 145.

Is the above calculation incorrect, or am I just a horrible counter... haha
Thanks!