Union and Intersection of Sets

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 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 set 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 

Study Science but not Mathematics

Study Mathematics but not Science

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.


3. 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.


4. 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.


5. 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 Mathematics is 140

FAQs (Frequently Asked Questions)

1. What Does the Symbol Mean?

In Mathematics, the intersection of two sets is the largest set that includes all the elements that are common to both the sets. In addition, the symbol for representing intersection of set is ∩.

Hence, symbolically we write intersection of two sets P and Q is P ∩ Q, which implies P intersection Q.

The intersection of two set P and Q is represented as ( P ∩ Q, x : x ∈  P and x ∈  Q).

2. What are the 4 Basic Operations of Sets?

The 4 basic operation of sets are:

  • Union of set

  • Intersection of set

  • Complement of set

  • Cartesian Product of set

3. What are the Properties of Intersection of Sets?

Some of the properties of intersection of sets are as follows:

  • Commutative Law - The union of two sets P and Q follows the commutative law i.e P ∩ Q = Q ∩ P.

  • Association Law = The intersection of operations always follows association law  i.e. if we have three set P, Q and R, the association law will be defined as (P ∩ Q) ∩ R = P ( Q  ∩ R)

  • Identity Law = The intersection of an empty set with any set P obtains the empty set itself i.e.

  • Distribution Law =  According to distribution law, if we have 3 sets then,

(P ∩ Q) ∩ R =  ( P  ∩ Q) ∪(P ∩  R).

  • Idempotent Law  = The intersection of any set P with itself obtains the set P i.e. P ∩ P = P.

  • Law of U = The intersection of universal set U with any set P obtains the set itself i.e.P ∩ U = P