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Math Help - Induction Proof.........

  1. #1
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    Induction Proof.........

    Prove by induction that the number of 2-subsets of an n-set A equals n(n-1)/2.
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  2. #2
    Junior Member roy_zhang's Avatar
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    Quote Originally Posted by st4rl1ight View Post
    Prove by induction that the number of 2-subsets of an n-set A equals n(n-1)/2.
    First please note that a 2-subset is a subset of a set on n elements containing exactly 2 elements, so the number of 2-subsets of an n-set is just \binom{n}{2}=\frac{n(n-1)}{2}. Now let's prove it by induction:

    Base Step: When n=1, it is trivial (we have \frac{1(1-1)}{2}=0 2-subset); when n=2, we have \frac{2(2-1)}{2}=1 2-subset.

    Inductive Step: Now lets assume when n=k, we have \frac{k(k-1)}{2} 2-subsets. Consider the case when n=k+1 (i.e. we add one more element into the set, so how many more 2-subsets will be introduced? The answer is k more 2-subsets are introduced. Think about this)
    So the total number of 2-subsets when n=k+1 is \frac{k(k-1)}{2}+k=\frac{(k+1)(k+1-1)}{2}, this competes the inductive step.

    Roy
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