Proving Set Theory Equation: Subsets of Size m = Subsets of Size n-m

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Homework Help Overview

The discussion revolves around proving a set theory equation related to the number of subsets of a set {1, ..., n}. The original poster seeks to establish that the number of subsets of size m is equal to the number of subsets of size n - m, for all n and m where 0 <= m <= n.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the proof and considers examining the powerset of the sets of size m and n - m. Some participants mention combinations and question their understanding of the concept. Others inquire about the implications of choosing m elements from the set.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some have suggested potential relationships between combinations, while others are seeking clarification on foundational concepts related to subsets and their sizes.

Contextual Notes

There appears to be a lack of familiarity with combinations among some participants, which may affect their ability to engage with the problem fully. The original poster's approach to finding a one-to-one correspondence between subsets is also under consideration.

TalonStriker
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Homework Statement



Prove that, for all n, for all m with 0 <= m <= n, the number of subsets of {1, . . . , n} of size m is the same as the number of subsets of {1, . . . , n} of size n − m.


Homework Equations


n/a


The Attempt at a Solution


My problem is that I don't know where to begin. I have a vague notion that I should somehow find the powerset of all the sets of size m and n - m and compare their sizes. But I don't really know where to start.
 
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Combinations?
 
C(n,m)=C(n,n-m) since order of elements in a set is not important.
 
EnumaElish said:
Combinations?

I am not familiar with combinations... what are they?

A google search doesn't net me any useful results.
 
Last edited:
I want to choose m things. Therefore, I don't choose how many things?
 
If A is a subset of S= {1, 2,... n} with m members, how many members does S\A have? Do you see an obvious one-to-one correspondence?
 

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