Intersection of Sets using Venn Diagram

Learn how to represent the intersection of sets using Venn diagram. The intersection set operations can be visualized from the diagrammatic representation of sets.

The rectangular region represents the universal set U and the circular regions the subsets A and B. The shaded portion represents the set name below the diagram.

Let A and B be the two sets. The intersection of A and B is the set of all those elements which belong to both A and B.

Now we will use the notation A ∩ B (which is read as ‘A intersection B’) to denote the intersection of set A and set B.

Thus, A ∩ B = {x : x ∈ A and x ∈ B}.

Clearly, x ∈ A ∩ B   

⇒ x ∈ A and x ∈ B

Therefore, the shaded portion in the adjoining figure represents  B.

Intersection of Sets using Venn Diagram

Thus, we conclude from the definition of intersection of sets that A ∩ B ⊆ A, A ∩ B ⊆ B.


From the above Venn diagram the following theorems are obvious:

(i) A ∩ A = A                        (Idempotent theorem) 

(ii) A ∩ U = A                       (Theorem of union) 

(iii) If A ⊆ B, then A ∩ B = A.

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

(v) A ∩ ϕ = ϕ                       (Theorem of ϕ) 

(vi) A ∩ A’ = ϕ                      (Theorem of ϕ) 

The symbols ⋃ and ∩ are often read as ‘cup’ and ‘cap’ respectively.

For two disjoint sets A and B, A ∩ B = ϕ.


Solved examples of intersection of sets using Venn diagram:

1. If A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}. Find A ∩ B using venn diagram.

Solution:

According to the given question we know, A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}

Now let’s draw the venn diagram to find A intersection B.

Examples of Intersection of Sets

Therefore, from the venn diagram we get A B = {1, 3}


2. From the adjoining figure find A intersection B.

Intersection using Venn Diagram

Solution:

According to the adjoining figure we get;

Set A = {m, p, q, r, s, t, u, v}

Set B = {m, n, o, p, q, i, j, k, g}

Therefore, A intersection B is the set of elements which belong to both set A and set B.

Thus, A ∩ B = {p, q, m}

Set Theory

Sets Theory

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






8th Grade Math Practice

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