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Union and Intersection of Sets

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Last updated date: 04th Mar 2024
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Union and Intersection

The union of two sets P and Q is represented by P ∪ Q. This is the set of all different elements that are included in P or Q. The symbol used to represent the union of set is ∪.


The intersection of two set P and Q is represented by P ∩ Q. This is the set of all different elements that are included in both P and Q. The symbol used to represent the intersection of set is  ∩ . We can say that the intersection of two given sets i.e. P and Q is the set that includes all the elements that are common to both  P and Q.

 

Example:

If P = { 1,3,5,7,9} and Q = { 2,3,5,7}

What are P ∪ Q, and P   Q

 

Solution:

P ∪ Q = { 1,2,3,5,7,9}

 P Q = { 3,5,7}

 

A great way of learning Union And Intersection of Sets is by using Venn diagrams. The venn diagram of union and intersection is discussed below.

 

Union and Intersection Venn Diagram

A venn diagram is a diagram that represents the relation between and among a finite group of sets. If we have two or more sets, we can construct a Venn diagram to represent the relationship among these sets as well as the cardinality of sets. Venn diagrams are helpful in representing relationships in statistics, probability, and many more.

 

Venn diagrams are specifically used in set operation as they give us visual information of the relationship involved. 

 

To learn union and intersection through Venn diagram, we will represent sets with circles as shown below:

 

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Now we will place the values in appropriate places.

 

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The union of set is any region including elements of either A or B

 

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The intersection of sets is any region including the elements of both A and B.

 

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Union of Set

The union of two sets P and Q is equivalent to the set of elements which are included in set P, in set Q, or in both the sets P and Q. This operation can b represented as

P ∪ Q = { a : a ∈ P or a ∈ Q}

Let us understand the union of set with an example say, set P {1,3,} and set Q = { 1,2,4} then,

P ∪ Q = { 1,2,3,4,5}

 

Venn Diagram of Union of Sets

Let us look at the Universal set U such that A and B are the subsets of this universal set. The union of two sets A and B is defined as the set of all elements that are included in set A or set B or both. The symbol ‘∪’ is used to represent the union of two sets.

 

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In the venn diagram given above, the yellow coloured portion denotes the union of both the sets A and B. Hence, the union of two sets A and B is represented by a set C, which is also considered as a subset of the universal set U such that set C includes all those elements that are either in Set A or set B or in both A and B i.e. A ∪ B = { x : x ∈ A  or x ∈ B }.

 

Intersection of Sets

The intersection of two sets A and B which are subsets of the universal set U, is the set that includes all those elements that are common to both A and B.

It is represented by the symbol ‘ ∩’. All those elements that are included in both set A and B denotes the intersection of A and B. Hence we can say that, A ∩ B = {  x : x  ∈ A and x ∈ B }.

For n sets i.e. A₁ ,A₂, A₃,....An, where all these sets are the subset of the universal set U, the intersection is the set of all the elements which are common to all these n sets.

Representing this periodically, the shaded portion in the Venn diagram given below denotes the intersection of two sets A and B.

 

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Intersection of Two Sets Representation

If X and Y are two sets, then the intersection of two sets is represented by 

X ∩ Y = n( X) + n(Y) - n( X ∩ Y)

Where n(X) is the cardinal number of set X, n(Y) is the cardinal number of set Y, n( X ,Y ) is the cardinal number of union of set X and Y.

To understand the concept of intersection of two sets clearly, let us consider an example.

If Set X = { 4,6,8,10,12 }, Set Y = { 3,6,9,12,15,18} and Set Z = { 1,2,3,4,5,6,7,8,9,10}. Find the Intersection of 

1.    Set X and Y

2.    Set Y and Z

3.    Set A and C


Solution:

  • Intersection of set X and Y is X ∩ Y

Set of all the elements which are common to both set X and Y is {6, 12}

  • Intersection of set Y and Z is Y ∩ Z

Set of all the elements which are common to both set Y and Z is {3,6,9}

  • Intersection of set X and Z is X ∩ Z

Set of all the elements which are common to both set X and Z is {4,6,8,10}

 

Cardinal Number of Set

The number of different elements included in a finite set is called its cardinal number of a set. The cardinal number of sets is represented as n(A) and read as ‘number of elements of the set’.

 

For example, 

Set X = { 2,4,5,9,15 } has 5 elements

Hence, the cardinal number of set X = 5. Hence, it is represented as n(x) = 5.

 

Difference between Union and Intersection of Set

Union of Set

Intersection of Set

The union of two sets X and Y is defined as the set of elements that are included either in the set X or set Y, or both X and Y.

The intersection of two sets X and Y is defined as the set of elements that belongs to both sets X and Y.

The symbol ∪ is used to represent the union of two sets.

The symbol ∩ is used to represent the intersection of two sets.

The union of set corresponds to the logical OR

The intersection of set corresponds to the logical And

It rejects the identical values from the set 

It is an associative operation which includes the common values from the set 

Example: If X = { 1,2,3} and Y = { 2, 3, 4}, then X ∪ Y = { 1,2,3,4}

Example: If X = { 1,2,3} and Y = { 2, 3, 4}, then X ∩ Y = { 2,3}

 

Union and Intersection Examples

1. If X = {Multiples of 3 between 1 and 20}, and Y = ( Odd Natural Numbers upto 14}. Determine the intersection of two given sets X and Y.

Solution: 

X = { Multiples of 3 between 1 and 20}

Hence, X = { 3,6,9,12,15,18}

Y = { even natural numbers upto 15}

Hence, Y = { 2,4,6,8,10,12,14}

Therefore, intersection of X and Y is the largest set including only those elements which are common to both the given sets X and Y.

Hence, X ∩ Y = { 6, 12 }

 

2. P = { 1,3,7,5} and Q = { 3,7,8,9}. Find the union of two sets P and Q.

Solution:

P ∪ Q = { 1,3,5,7,8,9} 

No elements are repeated in the union of two sets. The common elements i.e. 3 and 7 are considered only once. 

 

Union and Intersection of Sets Cardinal Number Practice Problems

1. Find the Union and Intersection of two sets P and Q Where Set P = { -29, -45, -10, -30, -3, -39, 24} and Set Q = { -46, 21 ,-8}. What is the Cardinal Number of P,Q, their union and intersection?

Solution:

Union = { -29, - 45, -10, - 30, - 3, - 39, 24, - 46, 21,- 8}

Intersection {}

Cardinal number of P = Number of elements in P = 7

Cardinal number of Q= Number of elements in Q = 3

Cardinal number of union of two sets = Number of total elements in both the sets = 10

Cardinal number of intersection of two sets= Number of elements in their intersection = 0 ( Null set).

 

2. There are a total number of 200 students in Class XI. Among them, 120 students study science, 50 students mathematics, and 30 students study both science and mathematics. Find the number of students who 

  1. Study science but not mathematics

  2. Study mathematics but not science

  3. Study science or mathematics

 

Solution:

The total number of students denotes the cardinal number of the universal set. Let x  represent the set of students studying Science and set Y represent the students studying Mathematics.

Therefore,

n(U) = 200

N(X) = 120

N(Y) = 50

N ( X ∩ Y) = 30

The venn diagram denotes the number of students studying both Science and Mathematics.

 

i. Number of students studying science but not mathematics

Here, we are required to find the difference of sets X and Y

n(X) = n( X - Y) + n (X ∩ Y)

n( X - Y) = n(X) - n(X ∩ Y)

n( X -Y) = 120 -30

= 90

Hence, the number of students who study Science but not Mathematics are 90.

 

ii. Number of students studying mathematics but not science.

Here, we are required to find the difference of sets Y and X

n(Y) = n( Y - X) + n (X ∩ Y)

n( Y - X) = n(Y) - n(X ∩ Y)

n( X -Y) = 50 -30

= 20

Hence, the number of students who study Mathematics but not Science are 20.

 

iii. Number of students who study science or mathematics

n( X∪ Y) = n(X) + n(Y) - (X ∩ Y)

n ( X ∪Y) = 120 + 50 - 30 = 140

Hence, the number of students who study Science or Math

 

Union And Intersection

A set can be defined as a collection of elements or items which can be mathematical like functions, numbers or it may not be mathematical. The usage of sets must have been older than the use of numbers itself. The number of animals in a herd can be counted with the stones in a sack without the actual members being counted. This notion goes up to infinity. For example, when we consider the set containing integers then the integers from 1 to 100 are finite but if we consider the whole set then it will be infinite. Sets are represented with many members together enclosed by brackets. When there are no members in the set then it is called a null or empty set.  The infinite sets are represented by a formula which gives the elements when used for the elements of the set.


The union function of two sets has all the elements or objects present in two sets or either of the two sets. It is represented by ⋃. The intersection function of two sets is when all the elements present in the both sets are present. It is represented as ⋂.


The complement function of intersection doesn’t contain everything that is present on the set. On the other hand, a universal set has all the elements we need in a set. A complement set can be said to be relative to the universal set.

FAQs on Union and Intersection of Sets

1.  What is the intersection of sets?

Intersection of two sets is the set which has all the elements that are found in both the sets given. It is given by the symbol ∩. When there are two sets, say X and Y then the intersection can be written as X ∩ Y which has all the elements present in both the sets i.e. the elements common to both X and Y sets. Say, the set X = {1, 3, 4, 5} and set Y = {2,3,5,6} then the intersection of both these sets (X ∩ Y) will be {3,5} i.e. both the elements that are present in the set X and Y.

2. What is a cardinal number?

Cardinal number can be defined as the set of all the elements that are present in the set. It is the total number that is present. For example, set X = {1,3,4,5} then the cardinal number which is represented as n(X) will be 4 as there are 4 numbers of elements present in the set. They can also be used for counting the intersection and union of elements and finding the total number of elements after these functions are performed on two sets. 

3. What are venn diagrams?

Venn diagrams can be defined as the diagrams we use to explain or represent the relationship between operations of sets. These diagrams are circular and each circle represents a set. When these circles overlap then it means that the sets that are taken for consideration have some relationship or they have common elements from the sets. On the other hand, when they do not overlap then it means that there are no common elements of the sets taken into consideration. 

4. What are universal sets?

Universal sets are made of two or more sets which are found to be the subsets of the universal set itself. It is denoted by U. it can also be defined as a set or a larger set which has all the elements of the related sets and has no repetition of the elements. Let us understand this with an example. Two sets, say X = {a,b,c} and B ={1,2,3,a,b} then the universal set which is related to these sets will be U = {1,2,3,a,b,c}.

5. What is the complement of a universal set?

Universal sets which are represented by U have sub-sets. For example, for a subset X of the universal set, the complement of it can be written as X’. In the complement set, the elements of the universal set are present and not the elements of the subset X. The universal set has all the elements of the subsets that are related to it but an empty set has no elements. The complement of such a universal set will be zero that has no elements and can be represented as ‘{}’ or by the symbol phi.