# 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

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.

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}

Set Theory