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

by dpesios
Tags: elements, nn1 or 2, subsets
 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.
HW Helper
P: 1,338
 Quote by dpesios * 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?
 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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