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 A ∩ B.

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.

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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.

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

2. From the adjoining figure find A intersection B.

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}

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● 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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