Subset

A subset is a set whose elements are also a part of another set. For instance, if all elements in the set A are a part of set B, then set A is a subset of set B. A subset is represented using the symbol ⊆, in set theory. For example, if A = {a, b, c} and B = {a, b, c, d, e, f}, then A is a subset of B.  

To find the total number of subsets of a set, we use the formula: 2ⁿ, where n is the number of elements in the set. For example, if A = {2, 4, 6}, then the number of subsets of a set A 2³ = 8. 
 

The subset and superset are types of sets. If A is a subset of B, then B is a superset of A. In this section, we will discuss the difference between a subset and a superset.
 

Subsets are classified into different types based on the number of elements they have. The main types of subsets are: 

• Proper Subset
• Improper Subset
• Singleton Subset

Proper Subset: A proper subset includes all sets except the set itself. For example, if A = {3, 6, 9}, then its proper subsets are {}, {3}, {6}, {9}, {3, 6}, {3, 9}, {6, 9}, but the set {3, 6, 9} is not a proper subset. If A is a proper subset of B, then it can be represented as A ⊂ B, where A ≠ B, which means elements in set A are a part of set B, but A does not have all elements of B. 

Improper Subset: The improper subset of a set is the set itself. For example, for the set {3, 6, 9}, the improper set is {3, 6, 9}. If A ⊆ B and A = B, then A is a subset of B, but not a proper subset. 

Singleton Subset: The subset of a set that contains only one element of the set is a singleton subset. For example, A = {5, 6, 7}, then the singleton subset will be {5}, {6}, {7}. 

In set theory, to represent a subset, we use symbols like ⊆ and ⊂. To represent the proper subset, we use the symbol ⊂, and to represent the improper subset, we use the symbol ⊆. The symbol ⊆ is read as a subset or equal to, and ⊂ is read as a subset of. 

A proper subset of a set is a subset that contains some elements, but not all, elements of the original set. It does not include the set itself. For example, if A = {2, 4, 6}, then its proper subset are: {}, {2}, {4}, {6}, {2, 4}, {4, 6}, and {2, 6}, but {2, 4, 6} is a subset but not a proper subset. A proper subset can be represented using the symbol ⊂, where A ⊂ B and A ≠ B. The number of proper subsets of a set can be calculated using the formula: 2n - 1, where n is the number of elements. 

The numbers, including positive integers, negative integers, fractions, and irrational numbers, are the real numbers. The subset of real numbers includes:

• Rational Numbers: The numbers in the form p/q, where p and q are integers and q ≠ 0, and include terminating and repeating decimals. 
• Integers: The numbers include all the whole numbers and integers, including zero. 
• Whole Numbers: The numbers include all positive numbers and zero.
• Natural Numbers: The numbers include all positive integers. 

Integers include positive and negative numbers and zero. It doesn't include the fractions and decimals. To represent the set of integers, we use the symbol Z. 
 

The power set of a set includes all the possible subsets of the set, including the empty set and the set itself. The power set of a set is represented by p(A), where A is the original set. 
For example, if A = {5, 10}, then the power set of A is denoted as p(A), 
p(A) = {{}, {5}, {10}, {5, 10}}

In set theory, a subset is a fundamental concept and is used in different fields such as computer science, biology, economics, etc. In this section, we will learn a few real-world applications of subsets. 

• In online shopping platforms, subsets are used to display products based on selected criteria. For example, if you are searching for a dress in white color, under $15, then the website will show only items based on the criteria.  

• In event planning and scheduling, subsets are used to represent particular combinations. 

• In biology, to model gene regulatory networks, we use subsets.

• In economics, we use subsets to divide the market into segments based on customer characteristics.