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Induction Proof

  1. Mar 14, 2008 #1
    Prove by induction that the number of 2-subsets of an n-set A equals n(n-1)/2.
     
  2. jcsd
  3. Mar 15, 2008 #2
    Not sure why you want to use induction to do this but here you go in a fast and loose manner:

    Assume it's true for a set of size n. Examine a set of size n+1. Notice this set of size n+1 is just an n-set with a new element added to it. Call that new element 'A'. Now what does a 2-subset of the n+1-set look like? Either it has A or it doesn't. Thus, using our inductive hypothesis, we see there are:

    [tex]\frac{n*(n-1)}{2} + n = \frac{(n+1)*n}{2}[/tex]

    2-subsets. And that's exactly what we want to see.
     
  4. Mar 16, 2008 #3
    you mean, considering a set of n elements the number of subsets with 2 elements obtained from that set is equal to n(n-1)/2?
     
  5. Mar 16, 2008 #4
    if so, it is easy to see that the first combine with n-1 elements, the second with n-2 elements, and so on
     
  6. Mar 17, 2008 #5
    Yes That gives a direct formula but what the poster needed was a step for an inductive proof which is what Rodigee gave since the new element (n+1) combines with the first n elements to form n more sets of two elements.
     
    Last edited: Mar 17, 2008
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