Set Theory: Types, Representation, Symbols, Formulas

Set Theory, in Mathematics, is a branch of mathematics which helps us understand a collection of objects, usually called, sets. These well-defined objects are also known as elements and could be of any kind and in any form.

• These may be alphabets, numbers, lines, shapes or even variables.
• For example, since the number of players found in a cricket team can only be 11 at one time, thus it is a finite set.
• Georg Cantor (1845-1918), a German mathematician, first gave the concept of ‘Theory of sets’ or ‘Set Theory’. 
• An element ‘a’ which belongs to a set A can also be represented as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.

Sets can be represented in a variety of methods, such as:

• Statement Form
• Roaster Form or Tabular Form Method
• Set Builder Method

Read Also: NCERT Solutions for Class 11 Maths Sets

Key Terms: Set, Associative Property, Distributive Property, Statement Form Method, Roster Form Method, Set Builder Form Method, Null Set, Finite Set, Infinite Set, Unit Set

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What is Set in Maths?

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A set is a well-organised collection of elements. It is usually denoted by a capital letter followed by curly braces. For example, if we say to make a set of all the vowels in the english alphabet, then it will be represented as follows,

⇒ S = {a, e, i, o, u}

Set can be simply defined as:

• The elements present in a finite set is known as a cardinal number.
• A finite set is a set which contains a fixed or a definite number of elements.
• A set is characterised by a capital letter, which is, A = {...}. 
• Elements, in this case, are expressed using a curly bracket in a set, A = {1, 2, 3, 4, 5}. 

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Representation of Sets 

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There are three basic methods devised to represent sets in mathematics, namely

• Statement Form Method
• Tabular Form Method or Roster Form Method
• Set Builder Form Method

Statement Form Method

In the Statement Form Method, all details regarding the set are given in words enclosed within two curly braces. For example, if we want to write a set of all the prime numbers less than 100, it will be written as, 

⇒ {prime numbers less than 100}

Another example of the same is:

A set of Odd Numbers is considered to be Less than 10.

Thus, in a Statement Form method, the elements can be shown as {1, 3, 5, 7, 9}.

Tabular or Roster form Method 

In this kind of set, the elements present inside the curly braces are separated from each other with the help of commas. For example, write a set of all vowels present in the English alphabet represented by the letter V.

⇒ V = {a, e, i, o, u}

Set Builder Form Method

In a Set Builder Form Method, every element has a common property. However, this property is not applicable to the objects which do not belong to the set.

For example, if i want to write a set A of all odd natural numbers

A = {x : x is a natural number and x = 2n + 1 for n ∈ W} 

Or, write P = {2, 4, 6, 8, 10}, writes this in set builder form

P = {x : x is a natural even number and 1 < x < 12}

Another example is when the set S has elements tat are even prime numbers. Thus, it can be represented as:

⇒ S = { x: x is an even prime number}

Here,

• x = symbol used to represent the element
• ‘:’ = ‘such that’
• ‘{}’ = ‘the set of all’

Set Detailed Video Explanation

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Types of Sets

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There are a variety of set types, including:

​Null Set or Empty Set

A set with absolutely no elements inside it is called an empty set or null set. The cardinality or count of elements of this set is 0. For example, name a month which consists of only two Mondays. We know that it is not possible, because Monday comes at least 4 times in a month.

Hence, it is represented as set A = {...}. It is denoted by the symbol Φ and is to be read as ‘phi’.

It is sometimes also known as a void set. 

Read More: Difference Between Variance and Standard Deviation

Singleton Set or Unit Set

For example, in set A, the name Monday appears in a week. 

Here, A = {1}

A set which consists of only one element is known as a unit set or a singleton set. The cardinality of set A in this scenario is 1. 

Finite set

A set which includes countable elements, or a definite number of elements, is known as a finite set. 

For example, name all the prime numbers from 1 to 20

A = {2, 3, 5, 7, 11, 13, 17, 19}

The cardinality of this set is 8 as it contains eight definite elements. 

Other example of a finite set,

S = { x | x ∈ N and 100 > x > 70 }, where N is Natural numbers

Infinite set

A set, the elements of which will not be finite, is called an infinite set. 

For example, S = { x | x ∈ N and x > 55 }

Now here, above 55 there are countless natural numbers present. The cycle will be a never ending one. Hence, an infinite set. 

Equal set

Two sets will be called equal sets, when both of them have the same elements as the other. 

For example, set A = { 2, 3, 5, 7, 9 }

Set B = { 2, 7, 9, 5, 3 }

Here, in the case given above, we will call both sets A and B as equal. This is because both have the same elements.

Equivalent set

Two sets are said to be equivalent when the cardinality of both are same. In simpler terms, it means that if the number of elements of set A is 5, then set B will also contain 5 elements. It is important to remember that the elements may not be identical. 

For example, Set A = { 2, 4, 8, 16, 32 }, cardinality = 5

Set B = { blue, yellow, green, red, pink }, cardinality = 5

Subset 

If set A contains elements, which belong to B as well, it is known as A is a subset of B. It is represented in the manner, A ⊆ B.

For example, 

If A = { 5, 10, 15 }

Then, Subsets of A= Φ, {5}, {10}, {15}, {5, 10}, {10, 15}, {15, 5}, {5, 10, 15}

Proper subset

A is said to be a proper subset of B when all elements of A are the elements of B. 

For example, Set A = {1, 2, 3, 4, 5}

Set B = {2, 4, 5}

Here, it can be concluded that, B is a proper subset of A, as all elements of B are contained in set A. We can write it as B ⊂ A.

Power set

It is the set of all the subsets for any given set. It is written as P(A).

For example, if a set consists of n number of elements, then the total number of subsets will be 2 to the power n or 2n elements. 

If set A = {5, 10, 15}, then power set of A will be,

P(A) = 23 = 8 elements in total

P(A) = Φ, {5}, {10}, {15}, {5, 10}, {10, 15}, {15, 5}, {5, 10, 15} = 8 elements

Universal set

A universal set is exactly its name. It considers things as a whole and is denoted by the letter U. 

For example, we can say that all the countries in the world may belong to a universal set. Every other set is a subset of a universal set. In a universal set, each element is unique and not repeated. 

For example, if,

Set A = {1, 4, 3, 5 }

Set B = { 2, 4, 5, 7 }

Set C = { 3, 7, 9, 11 }

The universal set, U = {1, 2, 3, 4, 5, 7, 9, 11}

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Operation of Sets 

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Set operations can be classified into two types:

• Union of sets
• Intersection of sets 

Union of Sets

The Union of sets X and Y is equivalent to the set of elements found in set X, set Y or both together. It can be expressed in terms of:

The union of sets calculates all the elements of the set as it relates to universal. No element is repeated. It is written as, U. 

For example, 

Set A = { 1, 4, 5, 10 }

Set B = { 2, 5, 10, 7 }

Then, A U B = {1, 2, 4, 5, 7, 10 }

Intersection of Sets

The Intersection of Sets refers to another operation of set theory. As opposed to the union of sets, intersection only takes the elements which are common to both sets. It is denoted as inverted U or ∩. 

For example,

• Set A = { white, yellow, orange, black, blue }
• Set B = { pink, blue, red, orange, white, brown }
• Then, A ∩B = { white, blue, orange}

Read More: ​Sets Important Questions

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Set Theory Symbols

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The symbols of set theory help us carry out different operations with numbers. There are many important symbols, let's discuss some of them and understand their function. 

Let us consider the universal set U = {1, 2, 5, 7, 9, 11, 13}

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Set Theory Formulas 

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Some important Set Theory Formulas include:

Commutative Property 

The Commutative Property explains that whatever values we put into the places of x and y, the product will not change. The intersection of sets satisfies this theory, as well as the union. 

It is denoted by, 

• A∩B = B∩A 
• A∪B = B∪A

Associative Property ​

The position of the parenthesis does not affect the result set in case of an Associative Property. 

It is denoted by,

• (A∩B)∩C = A∩(B∩C)
• (A∪B)∪C = A∪(B∪C)

Distributive Property 

Both the union, as well as the intersection of sets satisfy the Distributive Property of set theory. 

It is denoted by,

• A∩(B∩C) = (A∩B)∩(A∩C)
• A∪(B∪C) = (A∪B)∪(A∪C)

Law of Ø and ∪

It states the following,

• A ∩ Ø = Ø
• U ∩ A = A
• A ∪ Ø = A
• U ∪ A = U

Complement of a Set

In Complement of a set, we have:

U = { 2, 4, 6, 8, 10 }

A = { 2, 4, 6}

Then, the complement of set A, will denote all those elements which belong to the universal set but not A.

A’ = { 8, 10 }

The formula is given as, 

• A∩AC = U
• A∩AC = ∅

Idempotent Law​

It states the formula,

• A ∩ A = A
• A ∪ A = A

Read More: Data Sets

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Applications of Sets

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Set theory has a wide range of applications in real life. Starting from mathematics, topology, data structure to engineering, its usage has become very essential and dependable. It is also used by engineers to work with circuits. 

Sets can be used in many areas, including:

• Boolean algebra
• Statistics
• Probability
• Graphs
• Topology

Read More: Bayes Theorem Formula

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Things to Remember 

• Sets are a collection of objects. These objects are also known as ‘elements’. The elements may be of any nature like geometric shapes, numbers, symbols, alphabets, etc.
• A set is usually denoted by a capital letter and curly braces. For example, if A is a set consisting of prime numbers from 1 to 10, then it is written as, A = {2, 3, 5, 7}
• There are different types of sets like finite sets, infinite sets, equal sets, equivalent sets, null sets, singleton sets, universal sets, power sets, subsets, proper subsets, and more.
• There are numerous formulas or properties of set theory, including Commutative Property, Associative Property, Distributive Property, Idempotent Law, and more. 

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Previous Year Questions

• If A and B are non-empty sets such that… (KEAM)
• Let A and B be two sets then (A∪B)′… (BITSAT - 1990)
• Two finite sets have m and n elements. The total number…
• Which of the following sets is a finite set…
• For any two sets A and B, A-(A−B) equals… (WBJEE - 2009)
• In a certain town… [KEAM]
• In a class of … [KCET 2014]
• If A\B = {a, b}, B\A = {c, d} and A\B = {e, f}… [KEAM]
• If n(A) = 43, n(B) = 51… [KEAM]