A set is a collection of well defined objects and intersection is one of the operations performed on sets.
Example : 1, 3, 5, 7 and 9 are distinct objects when they are considered separately and collectively they form a single set of size five. Written as {1, 3, 5, 7, 9}. Intersection of sets is the set of elements which are common in all the sets. The symbol used to denote intersection is $\cap$. In this page we will learn intersection of two sets, intersection of three sets, intersection of open sets, intersection of convex sets and few examples on intersection of sets.

## Intersection of Two Sets

Intersection of two sets $A$ and $B$ is the set that contains all elements of $A$ that also belong to $B$ and no other elements.
It is the set of all objects that are members of both $A$ and $B$. Symbol for intersection is sometimes replaced by the word 'and' between two sets.

Given below is an image of intersection of two sets.

For two sets $A$ and $B$, intersection of $A$ and $B$ is written as A $\cap$ B.

Mathematically intersection of two sets A and B is defined as:

$A \cap B$ = {$x$ : $x$ $\in$ $A$ $\wedge$ $x$ $\in$ $B$}

Example: Find the intersection of $A$ and $B$.
$A$ = {2, 5, 7, 9, 23}
$B$ = {5, 8, 9, 55, 59}

Solution:

Given:
$A$ = {2, 5, 7, 9, 23}

$B$ = {5, 8, 9, 55, 59}

We need to find the common elements from the above given sets.
From above we can clearly see 5 and 9 are the common elements.

Therefore $A \cap B$ = {5, 9}

## Intersection of Three Sets

The intersection is the set that contains elements that belong to all the given sets at the same time.
Given three sets $A$, $B$ and $C$. We need to find the common elements among the given sets.

Given below is an image of intersection of three sets.

Example : Find the intersection of $A$, $B$ and $C$.
$A$ = {Apple, Mango, Banana}
$B$ = { Strawbeery, Mango, Peech}
$C$ =  { Kiwi, Grapes, Mango}

Solution:

Given that:
$A$ = {Apple, Mango, Banana}

$B$ = { Strawbeery, Mango, Peech}

$C$ =  {Kiwi, Grapes, Mango}

Common elements from the above given set is Mango.

Therefore $A \cap B \cap C$ = {Mango}

## Intersection of Convex Sets

Convex set contains line segments between any two points in the set. It is a straight line between any two points of the set and is also contained in the set.
If a and b are points in a vector space the points on the straight line between a and b is given by
$x$ = $\lambda$ $a$ + ( 1 - $\lambda$) b for all $\lambda$ from 0 to 1.
Statement : Intersection of two convex sets is also a convex set.

Proof: Consider $A$ and $B$ to be convex.

Let $C$ = $A \cap B$

We need to prove that $C$ is convex

For this if $x$ and $y$ are elements of $C$ and being a scalar in (0, 1) then

$ax$ + $(1 - a) y$ will be an element of $C$.

Since $x$ and $y$ are in $C$ $x$ and $y$ are in $A$ also

As $A$ is convex we see that

$ax$ + $( 1 - a) y$ is in $A$      ----- (1)

Similarly $x$ and $y$ are in $B$, $B$ being convex

$ax$ + $( 1 - a) y$ is in $B$   ------ (2)

From (1) and (2) $ax$ + $(1 - a) y$ is in $A$ and $B$.

Therefore, $ax$ + $(1 - a) y$ is in $C$.

Hence Proved.

## Intersection of Open Sets

Open sets are defined as those sets which contain an open ball around each of their points.
For the open sets, the corresponding theorems hold:

1) The intersection of two open sets is open.

2) The space E and the null set 0 are open.

3) The sum of any number of open sets is open.
Statement : Finite intersection of open sets is open.

Proof: Lets us consider a set A is open if for every $x \in A$, there exists some $\in> 0$ such that $B(x, \in)$ = $(y : y \in C, d(x,y) < \in)$

Let $X$ = $A \cap B$ then any $x \in X$ implies $x \in A$ and $x \in B$

Assume $X$ is not open

Since x is not open there $x \in X$ such that $x$ is on the boundary of $X$.

$x$ is on the boundary of either $A$ or $B$.

As $x$ has to be an interior point of both $A$ and $B$. Therefore all elements of $X$ are interior points.

Therefore $X$ is open.

## Intersection of Sets Examples

Example 1: Find the intersection of $A$ and $B$.

$A$ = {$x$ ; $x$ is a number bigger than 5 and smaller than 10}
$B$ = {$x$ ; $x$ is a positive number smaller than 8}

Solution:
$A$ = {6, 7, 8, 9}

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

$A \cap B$ = { 6, 7}

Example 2: Find the intersection of $A$, $B$ and $C$.

$A$ = { $x$ ; 10 < $x$ < 15}

$B$ = { $x$ ; 7 < $x$ < 8}

$C$=  { $x$ ; -6 $\leq$ $x$ < 3}

Solution: As there are no common element

So $A \cap B \cap C$ = {}

Example 3:

Consider
$A$ = {1, 3, 5, 7, 8, 9, 10, 11}

$B$ = {2, 4, 6, 8, 10, 12}

$C$ = {1, 3, 4, 6, 7, 8, 10 }

then find the  $A \cap B \cap C$.
Solution:

Given that:

$A$ = {1, 3, 5, 7, 8, 9, 10, 11}

$B$ = {2, 4, 6, 8, 10, 12}

$C$ = {1, 3, 4, 6, 7, 8, 10}.
Common elements from the above given sets are 8 and 10.

Hence, we get $A \cap B \cap C$ = {8, 10}.