Set Notation

Set notation is used to represent sets 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, empty set, and complement 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 (\(U\)):

• The universal set represents all the elements from the sets that are being considered for operations.
• The universal set is represented by the symbols \(U\). For example, a set A = {0, 2, 4, 6} and a set B = {5, 7, 9}, then the universal set is 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 in the set, it 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.
• The complement of a set A is represented as \(U\) - A.
• Let us consider a set A = {1, 2, 3, 4} and U = {1, 2, 3, 4, 5, 6, 7, 8}. The complement of A is A' = \(U\) - 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.
• When performing the union of sets, each element appears only once; duplicates are not included.
• 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. A ∩ B results in the set of elements common to both 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
• \(A Δ B = (A U B) - (A ∩ B)\)
• \(A Δ B = (A - B) U (B - A)\)
   

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)} 

Set-builder notation is a way of representing a set using a logical rule that is true for every element in the set. It can involve one or more variables and defines the condition that elements must satisfy to belong to the set.

A set can be represented in two ways:
 

1. Roster form: Listing all individual elements of the set.
    
2. Set-builder form: Representing the set using a statement, equation, or inequality that describes its elements.

Learn simple strategies to understand and work with sets effectively. These tips help students grasp symbols, subsets, complements, and visualize set operations clearly.

• Understand symbols like ∈, ⊆, ∪, ∩, and complement (').
   
• Identify the universal set 𝑈 before working with subsets and complements.
   
• Use Venn diagrams to visualize unions, intersections, and complements.
   
• Check carefully if elements belong to a set or if one set is a subset of another.
   
• Practice with real-life examples like fruits, numbers, or classroom groups.

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 use set notation to group vehicles, like F = {buses, trucks, cars} for four-wheelers and T = {bikes, bicycles, scooters} for two-wheelers, helping improve road safety and management.
 

Environmental Studies: There are many species on Earth, and set theory can help classify them into groups. For example, M = {mammals} and H = {herbivores}; their intersection M ∩ H represents herbivorous mammals, which helps scientists study specific animal categories.
 

Hospital Medical Records: Set notation is used in hospitals to identify patients with common conditions. For example, T = {p1, p2, p3} represents patients with thyroid issues, and L = {p1, p4, p5, p6} represents patients with low blood pressure. The intersection T ∩ L shows patients with both conditions, helping doctors provide better treatment.

E-commerce product filtering: Online shopping platforms use set notation to group and filter products. For example, if A = {products under ₹1000} and B = {electronics}, then A ∩ B gives all electronic items under ₹1000. This helps customers easily find products that meet multiple conditions.

School timetables: Schools use set notation to organize subjects and teachers. For example, Set M = {Math, Physics, Chemistry} and Set E = {English, Economics, History}. Using union and intersection helps in scheduling classes without overlapping subjects or teachers.