Understanding Subset Relationships in Set Theory

Understanding the relationship between sets, particularly subsets and equal sets, is fundamental to set theory. This article will explore why set B is considered a subset of set A, and why these sets are in fact equal. We will also discuss the significance of the order of elements within a set and the concept of proper subsets.

The Basics of Sets and Subsets

Before diving into the specific relationship between set A {12} and set B {21}, it is important to establish a few basic concepts in set theory. A set is a collection of distinct elements, and the order in which these elements are listed is not significant. For example, set A and set B listed above are fundamentally the same, just with the elements written in different orders. This is a key point that often leads to confusion.

Why Set B is a Subset of Set A

The statement that set B is not a subset of set A is based on a misunderstanding of the definition of a subset. In set theory, if every element of set B is also an element of set A, then set B is a subset of set A. Given the sets A {12} and B {21}, it is clear that 12 is an element of set A and 21 is an element of set B. Therefore, set B is a subset of set A, and vice versa.

Equal Sets and Their Properties

Two sets are considered equal if they have exactly the same elements. In this case, set A and set B are equal because they contain the same single element, regardless of the order in which it is listed. This is a fundamental aspect of set theory: the properties of a set are determined by its elements, not by the order in which they are written. The fact that set A {12} and set B {21} have the same single element means that they are equal sets. As stated, every set is a subset of itself. Therefore, set B is a subset of set A, and set A is a subset of set B.

Proper Subsets vs. Equal Sets

A proper subset of a set is a subset that contains not all the elements of the original set. For example, if set A had multiple elements, and set B was a subset of set A with fewer elements, then B would be a proper subset of A. However, in the case of set A {12} and set B {21}, since they are equal and contain the same element, they are not proper subsets of each other. Instead, they are improper subsets, meaning they are subsets of each other without containing all the elements of the original set.

Conclusion

To summarize, set B is indeed a subset of set A, and set A is a subset of set B due to the nature of equal sets. The order in which elements are listed does not matter; what matters is the presence of the same elements in both sets. Understanding these fundamental concepts is crucial for grasping more complex ideas in set theory and related fields of mathematics.

References

1. Bourbaki, N. (1960). Theory of sets. éditions Hermann.

2. Denlinger, C. R. (2011). Modern mathematics: an elementary approach. Jones Bartlett Learning.