Define subset of a set with an example. Also define proper subset and improper subset.

Introduction

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. A subset is a set that is a part of a larger set, which may possibly include all the elements of the larger set.

Definition of Subset

A subset is defined as a set A that is part of a set B if every element of A is also an element of B. This is denoted as A ⊆ B. The formal definition of a subset is as follows:

If A and B are sets and every element of A is also an element of B, then: A is a subset of B, denoted by A ⊆ B

For example, if we have a set A = {1, 2, 3} and a set B = {1, 2, 3, 4, 5}, then A is a subset of B because every element in A is also in B.

Types of Subsets

There are two types of subsets: proper and improper subsets.

Proper Subset

A proper subset is a subset that is not equal to the set it is contained in. In other words, if A is a proper subset of B, then there exists at least one element in B that is not in A. This is denoted as A ⊂ B. The formal definition of a proper subset is as follows:

If A and B are sets, A is a proper subset of B, denoted by A ⊂ B, if every element of A is also an element of B and there exists at least one element in B that is not in A.

For example, if we have a set A = {1, 2, 3} and a set B = {1, 2, 3, 4, 5}, then A is a proper subset of B because every element in A is also in B and there is at least one element in B (4 and 5) that is not in A.

Improper Subset

An improper subset is a subset that is equal to the set it is contained in. This means that every element in A is also in B and every element in B is also in A. This is denoted as A ⊆ B. The formal definition of an improper subset is as follows:

If A and B are sets, A is an improper subset of B, denoted by A ⊆ B, if every element of A is also an element of B and every element of B is also an element of A.

For example, if we have a set A = {1, 2, 3} and a set B = {1, 2, 3}, then A is an improper subset of B because every element in A is also in B and every element in B is also in A.

Comparison of Proper and Improper Subsets

	Proper Subset	Improper Subset
Definition	A is a proper subset of B if every element of A is also an element of B and there exists at least one element in B that is not in A	A is an improper subset of B if every element of A is also an element of B and every element of B is also an element of A
Notation	A ⊂ B	A ⊆ B
Example	If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, then A is a proper subset of B	If A = {1, 2, 3} and B = {1, 2, 3}, then A is an improper subset of B

Conclusion

Understanding the concept of subsets, including proper and improper subsets, is fundamental in the study of discrete structures. It forms the basis for many other concepts and principles in mathematics and computer science. By understanding these concepts, we can better understand the structure and properties of sets, which are fundamental to many areas of mathematics and computer science.

Summary

A subset is a set that is a part of a larger set, which may possibly include all the elements of the larger set. There are two types of subsets: proper and improper subsets.

Analogy

Think of a subset as a smaller group of people within a larger group. For example, if the larger group is a class of students, a subset could be a group of students who are interested in playing sports.