Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How many subsets are there in {{1,2,3},{4,5,6},{7,8,9}}. 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).

share|improve this question
1  
"How many subsets are there in ..." makes sense (to me) only if what follows in the sentences is a set. It's not clear in that notation what are the elements of that set. If we consider each triplet as an element of our set (call {1,2,3}=A {4,5,6}=B...) we have a set of 3 elements, which give us 8 subsets. This would be quite a confusing way of specifying a set, but I cannot think of something more reasonable. –  leonbloy Jun 2 '11 at 15:07
    
That was the way the question was phrased. What would make more sense? –  soandos Jun 2 '11 at 15:09
1  
@leonbloy: I don't see why it's confusing… writing a set as {A, B, C} is standard notation, and here A happens to be {1,2,3}, etc. This seems a straightforward question to me. –  ShreevatsaR Jun 2 '11 at 15:11
    
@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? –  soandos Jun 2 '11 at 15:14

2 Answers 2

up vote 4 down vote accepted

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.

share|improve this answer
    
And the elements of A are sub-sub-sets and so are not counted? –  soandos Jun 2 '11 at 15:31
    
@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. –  ShreevatsaR Jun 2 '11 at 15:38
    
Got it, thanks. –  soandos Jun 2 '11 at 15:42

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.

share|improve this answer
    
Why isn't every element also a subset? –  soandos Jun 2 '11 at 15:25

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.