Set Notation

to represent the set representations and the operations performed on them. The set notations are classified into two categories, they are:

• Set notation for set representations
• Set notation for set operations

For example, the set of the first five natural numbers is N = {1, 2, 3, 4, 5}. Here, N is the name of the set. Every element of the set is separated by commas and enclosed by curly braces { }.
 

There are different types of set notation, such as the universal set, the empty set, and the complement set of a set. Some special symbols, like subset and belongs to, are used to relate elements to the same set or other sets.

The table below comprises the set notations and their set representations.
 

Universal set (μ): The universal set represents all the elements from the sets that are being considered for operations. The universal set is represented by the symbols U or μ. For example, a set A = {0, 2, 4, 6} and a set B = {5, 7, 9}, then the universal set could be the set of all whole numbers relevant to the problem.  
 

Null Set (Ø): The empty set is a set that does not contain any elements inside it. The null set is represented by the symbol Ø. For example, the set of natural numbers that are less than 1 is Ø because natural numbers start from 1. 
 

Subset (⊂): If every element of the set A is also present in the set B, then we can say that A is a subset of B. It is represented as A ⊂ B (A is a subset of B). For example, a set A = {1, 2, 3} and a set B = { 1, 2, 3, 4, 6, 7}. As every element of A is present in the set B. So, A ⊂ B. A proper subset is denoted by ⊂, while a subset that may include equality is denoted by ⊆
 

Belongs to (∈): If a particular element is present in the set, then that element belongs to the set. If an element is not present in the set, then it does not belong to the set. For example, let us consider a set A = {2, 3, 4}. The number 2 is present in the set A, so we can write it as 2 ∈ A. 
 

Complement Of A Set: The complement of a set represents the remaining elements that are not present in the set. That means the complement of a particular set A is μ - A. Let us consider a set A = {1, 2, 3, 4} and μ = {1, 2, 3, 4, 5, 6, 7, 8}. The complement of A is A' = μ - A = {5, 6, 7, 8}.

To represent the operations performed on sets, we use set notation. The main operations include union, intersection, difference, and delta.

The table below shows the set notations and their set operations.
 

Union (U): The union of the two sets A and B can be written as A U B. The A U B includes all the elements present in both sets A and B. While performing the union sets, no duplicate elements are involved. Let us consider two sets A = { 1, 2, 3} and B = { 7, 8, 9}. The union of A and B is A U B = { 1, 2, 3, 7, 8, 9}.
 

Intersection (∩): The intersection of two sets A and B is represented as A ∩ B. The A ∩ B results in the set of common elements present in the sets A and B. Let us consider two sets X = {5, 6, 7, 8, 9} and Y = { 7, 8, 9}. The intersection of X and Y is X ∩ Y = {7, 8, 9}.
 

Difference (-): The difference of two sets A and B is represented by A - B. A – B contains elements that are in A but not in B. 
For example, let us consider the sets A and B, where A = {2, 3, 4} and B = { 3, 4, 5}. The A - B = {2}.

Delta (Δ): The delta of two sets A and B is denoted by A Δ B. The A Δ B results in the set of remaining elements after removing the common elements between the sets A and B. The delta of the sets A and B is A Δ B = (A U B) - (A ∩ B) or A Δ B = (A - B) U (B - A).
 

Cartesian Product (×): The Cartesian product of two sets A and B is denoted by A × B and is a set of ordered pairs (a, b), where a is from the set A and b is from the set B. Let us consider the set A = {a, b, c} and B = { 1, 2, 3}, then A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)} 

The concept of set notations is used in many real-life situations like educational institutions, banking, library book organizations, companies, and online platforms. Let us see some of the real-life examples of set notations.
 

Traffic and Vehicle Classification:

Traffic systems follow the classification of vehicles based on their features, like wheels, models, and colors. They classify the vehicles into sets. For example, four-wheeled vehicles can be kept in one set, F = {buses, trucks, cars}, and two-wheeled vehicles in another set, T = {bikes, bicycles, scooters}. This will help the traffic systems in road regulations, by which authorities can take action against people, and also enhance safety measures. 

 
Environmental Studies:

As we know, there are many species on Earth. To classify them into different groups, we can use set theory. For example, we have two sets, M = {mammals} and H = {herbivores}. The intersection of M and H, i.e., M ∩ H, gives the herbivorous mammals. This helps scientists study specific animal groups.
 

Hospital Medical Records:

The set notations are used to identify the common patients in the hospital. The patients in the hospital are categorized into different sets. For example, the set of patients having thyroid issues, T = { p1, p2, p3}, and the set of patients having low blood pressure, L = {p1, p4, p5, p6}. The intersection T ∩ L gives the list of people having common patients, which helps the doctors to give them better treatments.