Combinations in Set Theory: Finding Subsets Containing 1 but Excluding 3

Set theory is a fundamental branch of mathematics that deals with the properties and relations of well-defined collections of objects. Understanding how to manipulate and count subsets of a given set, particularly when specific conditions are imposed, is a critical skill. In this article, we will explore the problem of finding all the subsets of size three from a particular set, with the conditions that the subset must contain the element 1 but not the element 3.

Introduction to Set Theory and Subsets

Set theory, while seemingly abstract, has numerous real-world applications, from computer science to statistical analysis. One of the central concepts in set theory is that of a subset. A subset is a set whose elements are all members of another set. For example, if we have a set ( A {1, 2, 3, 4, 5, 6, 7} ), the subset ( B {1, 3, 5} ) is a part of ( A ).

Problem Statement and Solution

The specific problem at hand is to find all possible subsets of size three from the set ( A {1, 2, 3, 4, 5, 6, 7} ) that contain the element 1 but do not contain the element 3. To solve this problem, we can break it down into simpler steps:

Step 1: Selecting the Element 1

First, we note that every subset containing 1 will have 1 as one of its elements. Therefore, we can focus on the remaining elements of the set ( {2, 4, 5, 6, 7} ), excluding 3 when forming the subsets.

Step 2: Choosing Two More Elements

We need to find subsets of size three from the set ( {2, 4, 5, 6, 7} ) to complete the original set. The number of ways to choose 2 elements from a set of 5 is given by the combination formula ( C(n, k) frac{n!}{k! (n-k)!} ). Here, ( n 5 ) and ( k 2 ).

Using the combination formula:

( C(5, 2) frac{5!}{2! cdot 3!} )

( C(5, 2) frac{5 times 4 times 3 times 2 times 1}{2 times 1 times 3 times 2 times 1} frac{20}{2} 10 )

Thus, there are 10 possible subsets of size three that contain the element 1 and do not contain the element 3.

Examples of Such Subsets

Let's list a few examples to make the concept clearer. The subsets that include 1 and exclude 3 are:

1, 2, 4 1, 2, 5 1, 2, 6 1, 2, 7 1, 4, 5 1, 4, 6 1, 4, 7 1, 5, 6 1, 5, 7 1, 6, 7

Conclusion and Practical Applications

Understanding how to systematically count or list subsets under specific conditions is useful in various fields, from cryptography to data analysis. In the context of search engine optimization (SEO), knowing how to manipulate and describe subsets can enhance the clarity and precision of web content, making it more accessible to search engines and users.

SEO and Keywords

The importance of optimizing content with relevant keywords cannot be overstated. Choosing the right keywords can significantly improve the visibility of your content on the internet and attract more targeted traffic to your site. For this article, the primary keywords include subsets, combination, set theory, combinatorics.

References

For a deeper dive into set theory and combinatorics, consider checking out:

Concrete Mathematics: A Foundation for Computer Science by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Combinatorics and Graph Theory by John M. Harris, Jeffry L. Hirst, and Michael J. Mossinghoff.