count the number of subsets How many subsets are there in $\big\{\{1,2,3\},\{4,5,6\},\{7,8,9\}\big\}$. The answer I gave was three, but I was told that was incorrect. There were no other words in the question (pairwise, etc).I was told by someone that I should have considered all possible combinations of all the elements but that makes no sense to me (to do that, not that I don't know how).
 A: A set {A,B,C} has 8 subsets. The subsets are:


*

*$\emptyset$, the empty set

*{A}, here {1,2,3}

*{B}, here {4,5,6}

*{C}, here {7,8,9}

*{B,C}, here {{4,5,6},{7,8,9}}

*{A,C}, here {{1,2,3},{7,8,9}}

*{A,B}, here {{1,2,3},{4,5,6}}

*{A,B,C}, the set itself, here {{1,2,3},{4,5,6},{7,8,9}}


More generally, a set with $n$ elements has $2^n$ subsets. One way of seeing this is that for each of the $n$ element you have 2 choices: whether to include it in the subset or not.
A: There are 8 subsets. As leonbloy mentioned in his comment, the set given to you has 3 elements. The elements happen to be sets themselves, but that doesn't matter. To be explicit, if $a=\{1,2,3\}$, $b=\{4,5,6\}$, and $c=\{7,8,9\}$ then the set is $A=\{a,b,c\}$ and its subsets are $\emptyset$, $\{a\}=\{\{1,2,3\}\}$, $\{b\}=\{\{4,5,6\}\}$, $\{c\}$, $\{a,b\}$, $\{a,c\}$, $\{b,c\}$, and $\{a,b,c\}=A$.
This may sound confusing at the beginning as one tends to think of "types": there are elements, then sets, then "families of sets" if need be, and rarely anything else. But in some contexts it is actually a natural situation. In set theory, everything is a set, so the elements of any set are always sets. 
