Here you will learn what is the intersection of sets with definition and venn diagram and examples.

Let’s begin –

## What is the Intersection of Sets ?

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

The intersection of A and B is denoted by \(A \cap B\) (read as “A intersection B”)

Thus, \(A \cap B\) = {x : x \(\in\) A and x \(\in\) B}.

Clearly, x \(\in\) \(A \cap B\) \(\iff\) x \(\in\) A and x \(\in\) B.

In the given figure, the shaded region represents \(A \cap B\). This is the **venn diagram for intersection of sets**.

Evidently, \(A \cap B\) \(\subseteq\) A, \(A \cap B\) \(\subseteq\) B.

**Note** : If \(A_1\), \(A_2\), …. , \(A_n\) is a finite family of sets, then their intersection is denoted by \(A_1 \cap A_2 \cap ….. \cap A_n\).

**Example** : If A = {1, 2, 3, 4, 5} and B = {1, 3, 9, 12}, then \(A \cap B\) = {1, 3}.

**Example** : If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8} and C = {4, 6, 7, 8, 9, 10, 11}, then find \(A \cap B\) and \(A \cap B \cap C\).

**Solution** : \(A \cap B\) = {2, 4, 6}

\(\therefore\) \(A \cap B \cap C\) = {4, 6}

## Formula to Find Number of Elements in A Intersection B

If A, B and C are finite sets, and U be the finite universal set, then

n(\(A \cap B\)) = n(A) + n(B) – n(\(A \cup B\))

where, n(A) = number of elements in set A

n(B) = number of elements in set B

n(\(A \cup B\)) = number of elements in union of sets A and B

**Also Read** : Other Formulas and Operation of Sets

**Example** : If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(\(X \cup Y\)) = 38, then find n(\(X \cap Y\)).

**Solution** : By using the above formula,

n(\(X \cap Y\)) = n(X) + n(Y) – n(\(X \cup Y\))

\(\implies\) n(\(X \cap Y\)) = 17 + 23 – 38

\(\implies\) n(\(X \cap Y\)) = 2