Intersection of Sets: Symbol, Formula, Properties & Examples

Intersection of Sets is an operation on sets that lists the common elements present in two or more sets. A set is a well-defined group of objects in mathematics. 

• Set Operations are operations that involve combining different sets in order to create a new set with unique properties. 
• Intersection of Sets is the set that includes all the elements that are present in both sets. 
• It is denoted by the symbol "∩". 
• Intersection A and B (also written as A intersection B) lists all the elements that are shared by two sets A and B and are present in both sets.

Example: Consider Set A = {1,2,3,4,5} and Set B = {3,4,6,8}.

So, A ∩ B = {3,4}

Thus, the Intersection of Set A and Set B is {3,4}.

Read More: NCERT Solutions for Class 11 Maths Sets

Key Terms: Intersection of Sets, Sets, Venn Diagrams, Set Operations, Set Theory, Elements, Union of Sets

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What is Intersection of Sets?

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Intersection of Sets is the set of all elements which are common to all the given sets. 

• In set theory, the intersection of sets is defined as the set of all the common elements in Set A and Set B. 
• The symbol '∩' denotes the Intersection of Sets.
• For instance, A ∩ B is read as A intersection B.

Example of Intersection of Sets

If B is the set of the first five multiples of 4, and A is the set of even numbers less than 10, then the intersection of these two can be found as follows:

A = {2, 4, 6, 8}

B = {4, 8, 12, 16, 20}

4 and 8 are the elements that both A and B share.

Thus, A ∩ B = {4, 8}

Cardinal Number of Sets

Cardinal Number of a set refers to how many unique elements are present in a finite set. The cardinal number of sets is written as n(A), which can be read as "number of set elements."

For instance,

Set X = {2, 4, 5, 9, 15} 

Consequently, Set X's cardinal number is 5. Consequently, it is written as n(x) = 5.

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Intersection of Sets Symbol

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The set containing all the components that both A and B share is the intersection of the two sets A and B, which are subsets of the universal set U.

• It is symbolized by the character "∩". 
• Intersection of sets A and B is represented by all the elements that are part of both sets. 

A Intersection B Formula 

The formula for the intersection of sets is given as 

A ∩ B = { x : x ∈ A and x ∈ B }

It means x is an element of A ∩ B, if and only if x is an element of both Sets A and B. Therefore, the word “AND” represents the intersection of sets.

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Intersection of Sets Venn Diagram

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Venn diagrams are the pictorial representation of sets in which each set is represented as a circle. They make it simple to understand where sets intersect. Circles that overlap indicate that there is a connection between two or more sets and that they share components. The circles that don't overlap also don't have anything in common. 

Consider Set A = {1, 2, 3, 4,} and Set B = {3, 4, 6, 8}.

A ∩ B = {3, 4}

Intersection of Sets Venn Diagram is given as

Intersection of Sets Venn Diagram 

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Intersection of Two Sets

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• Intersection of Two Sets, say X and Y, is the largest set containing all the elements shared by X and Y. 
• Intersection of two sets can be an empty set, meaning that there are no elements in the intersection set, or it can be a set with at least one element. 
• If A and B are two sets with the relationship A ∩ B = φ, then A and B are referred to as disjoint sets. 
• Therefore, there are no elements at the point where A and B meet.

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Intersection of Three Sets

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If A, B, and C are three sets, then the set of all elements that are shared by A, B, and C is the intersection of these three sets. It can be represented as A ∩ B ∩ C. 

Consider three sets, 

• A = {6, 8, 10, 12, 14, 16}
• B = {9, 12, 15, 18, 21, 24} 
• C = {4, 8, 12, 16, 20, 24, 28}

These three sets can be expressed as intersecting at A ∩ B ∩ C. The only element that connects A, B, and C is 12.

As a result, A ∩ B ∩ C = {12}

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Intersection of Sets Properties

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The following is a list of some properties of the intersection operation:

(I) Commutative Law: A ∩ B = B ∩ 

Given, A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 5, 7}.

Now, A ∩ B = {1, 2, 3, 4, 5, 6} ∩ {2, 3, 5, 7} = {2, 3, 5}

B ∩ A = {2, 3, 5, 7} ∩ {1, 2, 3, 4, 5, 6} = {2, 3, 5}

As a result, A ∩ B = B ∩ A.

(II) Associative Law: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Given, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {5, 6, 7, 8 }.

Now,

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

(A ∩ B) ∩ C = {3, 4} ∩ {5, 6, 7, 8} = { } = φ

And

B ∩ C = {3, 4, 5, 6} ∩ {5, 6, 7, 8} = {5, 6}

A ∩ (B ∩ C) = {1, 2, 3, 4} ∩ {5, 6} = { } = φ

As a result, (A ∩ B) ∩ C = A ∩ (B ∩ C)

(III) Law of φ and U: φ ∩ A = φ, U ∩ A = A

Take into account φ = { } and A = 10, 11, 12.

φ ∩ A = { } ∩ {10, 11, 12} = { } = φ

Let A = 4, 8, 12, 16, 20 and U = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

U ∩ A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} ∩ {4, 8, 12, 16, 20} = {4, 8, 12, 16, 20} = A

(IV) Idempotent Law: A ∩ A = A

Consider: A = {a, b, c, d, e} such that A ∩ A = {a, b, c, d, e} ∩ {a, b, c, d, e} = {a, b, c, d, e} = A

(V) Distributive Law: A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Consider three sets, A = = {2, 4, 6, 8}, B = {2, 3, 5, 7} and C = {3, 4, 5, 6}.

B U C = {2, 3, 5, 7} U {3, 4, 5, 6} = {2, 3, 4, 5, 6, 7}

A ∩ (B U C) = {2, 4, 6, 8} ∩ {2, 3, 4, 5, 6, 7} = {2, 4, 6}

A ∩ B = {2, 4, 6, 8} ∩ {2, 3, 5, 7} = {2}

A ∩ C = {2, 4, 6, 8} ∩ {3, 4, 5, 6} = {4, 6}

(A ∩ B) U (A ∩ C) = {2} U {4, 6} = {2, 4, 6}

Thus, A ∩ (B U C) = (A ∩ B) U (A ∩ C)

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Intersection of Sets Examples

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Here are a few examples on Intersection of Sets for a better understanding of the concept: 

Union and Intersection of Sets

Union of two sets A and B refer to the set of all those elements which are either in A or in B. On the other hand, the intersection of two sets A and B is the set of all elements which are common. The formulas for the union and intersection of given sets based on the cardinality of sets are as follows: 

If A and B are finite sets such that A ∩ B = φ, thus, 

n (A ∪ B) = n (A) + n (B).

If A ∩ B ≠ φ, then

n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

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Things to Remember

• Intersection of Sets is the largest set containing all the elements common in two or more sets.
• If A and B are two sets, then the set of all elements that are shared by sets A and B is the intersection of sets.
• It is denoted by the symbol “∩” and is written as ‘A ∩ B’.
• Intersection of Sets Formula is given as A ∩ B = {x : x ∈ A and x ∈ B}.
• Union of Sets refers to the set of elements that are a part of either of the given sets.