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 $A$ 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.
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}
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.
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}
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.
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.
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}.