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A set A of n elements has n(n-1)/2 subsets of 2 elements

by dpesios
Tags: elements, nn1 or 2, subsets
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dpesios
#1
Dec11-11, 12:59 PM
P: 10
I would very much like some help to the following problem.

1. The problem statement, all variables and given/known data

Using mathematical induction, prove that a finite set A of n elements has n(n-1)/2 subsets of two elements.

3. The attempt at a solution

* Base step n=2: 2(2-1)/2= 1 subset of two elements.
* Inductive step: assuming the statement holds for n=k, that is a set A of k elements has k(k-1)/2 (hypothesis)
We want to show that it also holds for n=k+1, that is a set A of k+1 has (k+1)(k+1-1)/2 elements.
How can we infer from the hypothesis ??? I have no idea ...

I have an engineering background so be as descriptive as you can.
Thank you in advance.
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eumyang
#2
Dec11-11, 01:26 PM
HW Helper
P: 1,347
Quote Quote by dpesios View Post
* Base step n=2: 2(2-1)/2= 1 subset of two elements.
* Inductive step: assuming the statement holds for n=k, that is a set A of k elements has k(k-1)/2 (hypothesis)
We want to show that it also holds for n=k+1, that is a set A of k+1 has (k+1)(k+1-1)/2 elements.
A set A of k elements has k(k-1)/2 subsets of two elements, as you said.
Suppose you add a new element to set A. How many new subsets can be created where one of the elements of these subsets is the new element?
dpesios
#3
Dec11-11, 01:46 PM
P: 10
if we add an element to the set which previously had k elements (that is now has k+1 elements) the new subsets that include the new element will be :

(k+1)k/2 - k(k-1)/2 = k

So, how can we argue that this will solve the problem ?


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