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

In summary, using mathematical induction, we can prove that a finite set A of n elements has n(n-1)/2 subsets of two elements. The base step n=2 shows that there is 1 subset of two elements. The inductive step assumes that the statement holds for n=k and shows that it also holds for n=k+1. The addition of a new element to the set creates k new subsets, proving that the statement holds for n=k+1.
  • #1
dpesios
10
0
I would very much like some help to the following problem.

Homework Statement



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

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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  • #2
dpesios said:
* 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?
 
  • #3
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 ?
 

1. What does the formula n(n-1)/2 represent in the context of a set with n elements?

The formula n(n-1)/2 represents the total number of subsets of 2 elements that can be formed from a set with n elements. It is also known as the combination formula, where n represents the total number of items and 2 represents the number of items chosen at a time.

2. How is the formula n(n-1)/2 derived for subsets of 2 elements?

The formula n(n-1)/2 is derived from the concept of combinations in mathematics. It represents the total number of ways to choose 2 elements from a set of n elements without considering the order of the elements. It is also equal to nC2 or n choose 2.

3. Can the formula n(n-1)/2 be used for sets with any number of elements?

Yes, the formula n(n-1)/2 can be used for sets with any number of elements. It is a general formula for finding the number of subsets of 2 elements in a set with n elements.

4. How is the formula n(n-1)/2 relevant in real-life applications?

The formula n(n-1)/2 is relevant in various fields, including statistics, probability, and data analysis. It is used to calculate the total number of combinations or possibilities in a given scenario, such as choosing a team of 2 members from a group of n people.

5. Are there any other formulas for finding the number of subsets of 2 elements in a set?

Yes, there are other formulas for finding the number of subsets of 2 elements in a set, such as n!/2!(n-2)! and (n^2-n)/2. However, the formula n(n-1)/2 is the most commonly used and efficient formula for this purpose.

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