2
$\begingroup$

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).

  • 0
    @ShreevatsaR I don recall if there were letter there or numbers, but I assume that that also would not matter. Why is my answer wrong?2011-06-02

2 Answers 2

4

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.

  • 1
    @soandos: The elements of A are just elements of A. There exist *subsets* of A like {1,2} or {1}, but they are subsets of A and not subsets of the set {A,B,C}. From the point of view of a set {A,B,C}, it doesn't matter what A, B, C, are; its subsets are exactly those sets in which element is either A, B, or C — namely, the 8 mentioned above. For instance a strange set like {{1,2,{3,4,5}}, "hello", 42} will still have exactly 8 subsets.2011-06-02
2

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.

  • 0
    Why isn't every element also a subset?2011-06-02