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Principal Operations on Sets include Intersection, Union, Difference of Sets, Complement of a Set, etc. To Visualise Operations of Sets we use Venn Diagrams. Difference Operation in Set Theory is a fundamental and important operation along with Union, Intersection Operations. Get to know about the Difference of Sets Definition, and how to find Difference of Sets, Difference of Sets Diagrammatic Representation in the further modules.

Difference of Sets Definition

The difference between Sets A and B is the Set of elements present in A but not in B. It is represented as A -B. The region shaded in orange denotes A -B and the one shaded in violet represents the difference between B and A i.e. B-A.

How to find the Difference of Sets using the Venn Diagram?

Let us consider Two Sets A and B that are Subsets of Universal Set U.

To find the difference between Sets A and B simply write the elements of A and take away the elements that are also present in Set B.

The difference of sets A, B is represented as such

A – B = {x : x ∈ A and x ∉ B}

A-B is the set of all the elements that are present in Set A but don’t belong to Set B.

A – B = {x : x ∈ A and x ∉ B} or A – B = {x ∈ A : x ∉ B}

Thus, we can say that x ∈ A – B

⇒ x ∈ A and x ∉ B

The Same Logic applies to B-A i.e. it contains all the elements that are included in Set B but don’t belong to Set A.

We need to be cautious about the way we compute the difference of sets. Since A-B is not the same as B-A and order does matter in the Sets Difference. Thus, we can say that Difference of Sets Operation isn’t commutative.

Identities involving Difference of Sets

There are quite a few operations that include Difference and Complement of Sets. We have stated some of the important identities related to the Difference of Sets and they may include operations such as Union, Intersection in between. They are in the following fashion

• A – A =∅
• A – ∅ = A
• ∅ – A = ∅
• A – U = ∅
• (A^C)^C = A
• DeMorgan’s Law I: (A ∩ B)^C = A^C ∪ B^C
• DeMorgan’s Law II: (A ∪ B)^C = A^C ∩ B^C

Solved Examples on Difference of Sets

1. If A = {1, 2, 3, 4, 5, 6, 7, 8, 9} B = {2, 4, 6, 8, 10, 12, 14, 16, 18}. Find Difference of Sets A-B and represent the Same using Venn Diagram?

Solution:

Given Sets A = {1, 2, 3, 4, 5, 6, 7, 8, 9} B = {2, 4, 6, 8, 10, 12, 14, 16, 18}

A-B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8, 10, 12, 14, 16, 18}

= {1, 3, 5, 7, 9}

A-B is the Set of Elements that are present in Set A and doesn’t belong to Set B.

Therefore, A-B = {1, 3, 5, 7, 9}

2. If A = {1, 2, 3, 4} B = { 3, 4, 5, 6}. Find Difference of Sets A and B using Venn Diagram?

Solution:

A = {1, 2, 3, 4} B = { 3, 4, 5, 6}

A-B = { 1, 2, 3, 4} – { 3, 4, 5, 6}

A-B = {1, 2}

A-B denotes the Elements in Set A but doesn’t belong to Set B. The Difference of Sets A-B is shaded for your reference.

3.  Let us consider Set A = {blue, green, red} & Set B = {red, orange, yellow}. Find the Difference of Sets A and B?

Solution:

Given Set A = {blue, green, red} & Set B = {red, orange, yellow} are

A-B = {blue, green, red} – {red, orange, yellow}

= {blue, green}

A-B denotes the colors that belong to Set A and don’t belong to Set B. A-B is shaded so that you can understand easily.

Disjoint Sets are the Sets whose intersection with each other results in a Null Set. In Set Theory if two or more sets have no common elements between them then the Intersection of Sets is an Empty Set or Null Set. This kind of Sets is Called Disjoint Sets. Go through the entire article to know about Disjoint Sets Definition, Disjoint Sets using Venn Diagram, Pairwise Disjoint Sets, etc.

Disjoint Sets Definition

Two Sets are said to be disjoint if they have no common elements between them. If elements in two sets are common then they are said to be Non-Disjoint Sets. Condition for Disjointness is just that intersection of the entire collection needs to empty.

For Example, A = { 4, 5, 6} B = { 7, 8, 9} then A and B are said to be Disjoint Sets since they have no common elements between them. Some Sets can have null set as Intersection without being Disjoint.

Disjoint Sets using Venn Diagram

Two sets A and B are disjoint sets if the intersection of two sets A and B is either a null set or an empty set. In other words, we can say the Intersection of Sets is Empty.

i.e. A ∩ B = ϕ

Pairwise Disjoint Sets

Definition of Disjoint Sets can be proceeded to any group of sets. Collection of Sets is said to be pairwise disjoint if it has any two sets disjoint in the collection. These are also called as Mutually Disjoint Sets.

Consider P is a set of any collection of Sets and A and B. i.e. A, B ∈ P. Then, P is known as pairwise disjoint if and only if A ≠ B. Therefore, A ∩ B = ϕ.

Example:

{ {7}, {3, 4}, {5, 6, 8} }

Solved Examples on Disjoint Sets Venn Diagrams

1. Determine whether the following Venn Diagram represents Disjoint Sets or not?

Solution:

S = {2, 4, 6 8} T = {1, 3, 5, 7}

Sets S, T doesn’t have any common elements between them. Thus, Sets S, T are said to be Disjoint.

2. If A = { 1, 2, 3, 4, 5, 6, 7, 8, 9} B = {14, 15, 16, 17, 18, 19}. Check whether the following are Disjoint Sets are not?

Solution:

Given Sets are A = { 1, 2, 3, 4, 5, 6, 7, 8, 9} B = {14, 15, 16, 17, 18, 19}

Sets A, B doesn’t have any common elements between them. Thus, Sets A, B are said to be Disjoint.

Set Operations are performed on two or more sets to obtain the combination of elements based on the Operation. There are three major types of Operations performed on Sets like Intersection, Union, Difference in Set Theory. Check out Representation of Intersection of Sets using Venn Diagram, Properties of Intersection of Sets, Solved Examples in the later modules.

Intersection of Sets Definition

Intersection of Sets A and B is the Set that includes all the Elements that are Common to Sets A and B. Intersection is represented using the symbol ‘∩’. All the elements that belong to both A and B denote the Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

If you have n sets i.e. A₁, A₂, A₃…..A_n all are Subsets of Universal Set U the intersection is the set of elements that are in common to n sets.

Intersection of Sets Venn Diagram

Consider Two Sets A and B and their Intersection is depicted pictorially using the following Venn Diagram. A, B are subsets in the Universal Set. Intersection of Sets is all those elements that belong to both the Sets A and B. Shaded Portion denotes the Intersection of Sets A and B. Intersection of Sets A and B is represented as A ∩ B and is read as A Intersection B or Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}.

Clearly, x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

Thus, from the Definition of Intersection, we can conclude that A ∩ B ⊆ A, A ∩ B ⊆ B.

Properties of Intersection of Sets

(i) A ∩ A = A (Idempotent theorem)

(ii)  A ∩ B = B ∩ A (Commutative theorem)

(iii) A ∩ U = A (Theorem of Union)

(iv) A ∩ ϕ = ϕ (Theorem of ϕ)

(v) A ∩ A’ = ϕ (Theorem of ϕ)

(vi) If A ⊆ B, then A ∩ B = A.

Solved Examples on Intersection of Sets using Venn Diagram

1.  If A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}. Find A ∩ B using the Venn Diagram?

Solution:

Given Sets are A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}

Draw the Venn Diagram for the given Sets and then find the Intersection of Sets.

Intersection is nothing but the elements that are common in both Sets A and B.

A ∩ B = { b, e, h}

2. If C = { 3, 5, 7} D = { 7, 9, 11}. Find C ∩ D using Venn Diagram?

Solution:

Given Sets are C = { 3, 5, 7} D = { 7, 9, 11}

Let us represent the given sets in the diagrammatic representation of sets

Intersection is nothing but the elements that are common in both Sets C and D.

C ∩ D = {7}

There are certain basic operations that can be performed on Sets. Similar to Addition, Subtraction, Multiplication Operations which we come across in Maths we have Union, Intersection, Difference in Set Theory. These operations are performed on two or more sets to result in a New Set depending on the Operation Performed. In the case of Union of Sets, all the elements are included in the result whereas in Intersection only common elements between the Sets are included.

At times you might get confused between Union of Sets and Universal Set. There is a slight difference between Union and Universal Set. Union of two or more sets is an Operation performed on them and includes the elements that are in both sets. On the other hand, Universal Set itself is a Set and has all the elements of other sets along with its set.

Union of Sets Definition

Union of Two Sets A and B is the set of elements that are present in Set A, Set B, or in both Sets A and B. The Union of Sets Operation can be denoted as such

A ∪ B = {a: a ∈ A or a ∈ B}

For instance Let A = { 3, 4, 5} B = { 4, 5, 6, 7} then A U B = { 3, 4, 5, 6, 7}

Go through the further sections to know How to represent the Union of Sets using Venn Diagrams.

Union of Sets Venn Diagram

Let us dive deep into the article to know about the representation of Union of Sets using the Venn Diagram. You can visualize the Operation of Sets with the Diagrammatic Representation.

Consider a Universal Set U in which you have the Subsets A and B. Union of Two Sets A and B is nothing but the elements in Set A and B or both the elements in A and B together. The Union of Sets is denoted by the symbol U.

A U B is the Union of Sets A and B. It is read as A Union B or Union of A and B. The Notation representing the Union of Sets A and B is given as follows

A U B = {x : x ∈ A or x ∈ B}.

It is evident that, x ∈ A U B

⇒ x ∈ A or x ∈ B

In the same way, if x ∉ A U B

⇒ x ∉ A or x ∉ B

Thus, from the definition of Union of Sets, we can say that A ⊆ A U B, B ⊆ A U B.

From the above Venn diagram, we can infer certain theorems

(i) A ∪ A = A (Idempotent theorem)

(ii) If A ⊆ B, then A ⋃ B = B

(iii) A ⋃ U = U (Theorem of ⋃) U is the universal set.

(iv) A ⋃ A’ = U (Theorem of ⋃) U is the universal set.

(v) A ∪ ϕ = A (Theorem of identity element, is the identity of ∪)

(vi) A ∪ B = B ∪ A (Commutative theorem)

Union of Sets Venn Diagram Examples with Solutions

1. If A = { 2, 5, 9, 15, 19} B = { 8, 9, 10, 13, 15, 17}. Find A U B using Venn Diagram?

Solution:

Given Sets are A = { 2, 5, 9, 15, 19} B = { 8, 9, 10, 13, 15, 17}

Let us draw Venn Diagram for the given questions considering the given sets.

A U  B is clearly {2, 5, 8, 9, 10, 13, 15, 17, 19} all the elements in both the Sets.

A U B = {2, 5, 8, 9, 10, 13, 15, 17, 19}

2. If P = { 3, 6, 9, 12, 15, 18} Q = { 2, 4, 6, 8, 10, 12, 14, 16, 18}, Find P U Q and the represent the same using Venn Diagram?

Solution:

Given Sets are P = { 3, 6, 9, 12, 15, 18} Q = { 2, 4, 6, 8, 10, 12, 14, 16, 18}

On Drawing the Venn Diagram for the given sets we have the P U Q as under

It is clearly evident from the Venn Diagram that P U Q = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

Thus, P U Q = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

Set Theory is a branch of mathematics and is a collection of objects known as numbers or elements of the set. Set theory is a vital topic and lays stronger basics for the rest of the Mathematics. You can learn about the axioms that are essential for learning the concepts of mathematics that are built with it. For instance, Element a belongs to Set A can be denoted by a ∈ A and a ∉ A represents the element a doesn’t belong to Set A.

{ 3, 4, 5} is an Example of Set. In this article of Introduction to Set Theory, you will find Representation of Sets in different forms such as Statement Form, Roster Form, and Set Builder Form, Types of Sets, Cardinal Number of a Set, Subsets, Operations on Sets, etc.

• Representation of a Set
• Types of Sets
• Finite Sets and Infinite Sets
• Power Set
• Problems on Union of Sets
• Problems on Intersection of Sets
• Difference of two Sets
• Complement of a Set
• Problems on Complement of a Set
• Problems on Operation on Sets
• Word Problems on Sets
• Venn Diagrams in Different Situations
• Relationship in Sets using Venn Diagram
• Union of Sets using Venn Diagram
• Intersection of Sets using Venn Diagram
• Disjoint of Sets using Venn Diagram
• Difference of Sets using Venn Diagram
• Examples on Venn Diagram

Set Definition

Set can be defined as a collection of elements enclosed within curly brackets. In other words, we can describe the Set as a Collection of Distinct Objects or Elements. These Elements of the Set can be organized into smaller sets and they are called the Subsets. Order isn’t that important in Sets and { 1, 2, 4} is the same as { 4,2, 1}.

Examples of Sets

• Odd Numbers less than 20, i.e., 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
• Prime Factors of 15 are 3, 5
• Types of Triangles depending on Sides: Equilateral, Isosceles, Scalene
• Top two surgeons in India
• 10 Famous Engineers of the Society.

Among the Examples listed the first three are well-defined collections of elements whereas the rest aren’t.

Important Sets used in Mathematics

N: Set of all natural numbers = {1, 2, 3, 4, …..}

Q: Set of all rational numbers

R: Set of all real numbers

W: Set of all whole numbers

Z: Set of all integers = {….., -3, -2, -1, 0, 1, 2, 3, …..}

Z+: Set of all positive integers

Representation of a Set

Set can be denoted using three common forms. They are given along the following lines by taking enough examples

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

Statement Form: In this representation, elements of the set are given with a well-defined description. You can see the following examples for an idea

Example:

Consonants of the Alphabet

Set of Natural Numbers less than 20 and more than 5.

Roaster Form or Tabular Form: In Roaster Form, elements of the set are enclosed within a pair of brackets and separated by commas.

Example:

N is a set of Natural Numbers less than 7 { 1, 2, 3, 4, 5, 6}

Set of Vowels in Alphabet = { a, e, i, o, u}

Set Builder Form: In this representation, Set is given by a Property that the members need to satisfy.

{x: x is an odd number divisible by 3 and less than 10}

{x: x is a whole number less than 5}

Size of a Set

At times, we are curious to know the number of elements in the set. This is called cardinality or size of the set. In general, the Cardinality of the Set A is given by |A| and can be either finite or infinite.

Types of Sets

Sets are classified into many kinds. Some of them Finite Set, Infinite Set,  Subset, Proper Set, Universal Set, Empty Set, Singleton Set, etc.

Finite Set: A Set containing a finite number of elements is called Finite Set. Empty Sets come under the Category of Finite Sets. If at all the Finite Set is Non-Empty then they are called Non- Empty Finite Sets.

Example: A = {x: x is the first month in a year}; Set A will have 31 elements.

Infinite Set: In Contrast to the finite set if the set has infinite elements then it is called Infinite Set.

Example: A = {x : x is an integer}; There are infinite integers. Hence, A is an infinite set.

Power Set: Power Set of A is the set that contains all the subsets of Set A. It is represented as P(A).

Example:  If set A = {-5,7,6}, then power set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Sub Set: If Set A contains the elements that are in Set B as well then Set A is said to be the Subset of Set B.

Example:

If set A = {-5,7,6}, then Sub Set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Universal Set:

This is the base for all the other sets formed. Based on the Context universal set is decided and it can be either finite or infinite. All the other Sets are Subsets of Universal Set and is given by U.

Example: Set of Natural Numbers is a Universal Set of Integers, Real Numbers.

Empty Set: 

There will be no elements in the set and is represented by the symbol ϕ or {}. The other names of Empty Set are Null Set or Void Set.

Example: S = { x | x ∈ N and 9 < x < 10 } = ∅

Singleton Set:

If a Set contains only one element then it is called a Singleton Set.

Example: A = {x : x is an odd prime number}

Operations on Sets

Consider Two different sets A and B, they are several operations that are frequently used

Union: Union Operation is given by the symbol U. Set A U B denotes the union between Sets A and B. It is read as A union B or Union of A and B. It is defined as the Set that contains all the elements belonging to either of the Sets.

Intersection: Intersection Operation is represented by the symbol ∩. Set A ∩ B is read as A Intersection B or Intersection of A and B. A ∩ B is defined in general as a set that contains all the elements that belong to both A and B.

Complement: Usually, the Complement of Set A is represented as A^c or A^‘ or ~A. The Complement of Set A contains all the elements that are not in Set A.

Power Set: The power set is the set of all possible subsets of S. It is denoted by P(S). Remember that Empty Set and the Set itself also comes under the Power Set. The Cardinality of the Power Set is 2ⁿ in which n is the number of elements of the set.

Cartesian Product: Consider A and B to be Two Sets. The Cartesian Product of the two sets is given by AxB i.e. the set containing all the ordered pairs (a, b) where a belong to Set A, b belongs to Set B.

Representation of Cartesian Product A × B = {(a, b) | a ∈ A ∧ b ∈ B}.

The cardinality of AxB is N*M where N is the cardinality of A and M is the Cardinality of B. Remember that AxB is not the same as BxA.