hi Bharath 
read below article for understand how to get;set; property work

This excerpt is taken from Learning Apex Programming, by Matt Kaufman.
In order for an Apex class to function as a controller for a Visualforce page, the variables and methods in the class need to use the correct syntax. The variables in the class are made accessible to the page through the use of getter and setter methods.

A getter method is a nonstatic method that has no input parameters and returns an object. Visualforce pages use getter methods to get objects from your custom controller. A setter method is a nonstatic method that has only one input parameter and does not return anything. Visualforce pages use setter methods to set the value of an object in your custom controller. Let's take a look at some getter and setter methods:

//First declare your variable
Account myAccount;
//Here is the getter
public Account getMyAccount(){

//We don't want to return a null value
if ( myAccount == null ){
myAccount = new Account();
}
return myAccount;
}
//Here is the setter
public void setMyAccount(Account a){
myAccount = a;
}

Pretty neat, right? Just by naming our method getX or setX and using the right signature, they are automatically accessible by our Visualforce page. This just gets better; our getter and setter are using a whole record and our page has access to all of the fields on that record. We can even use a collection of records and have access to all of the records in the collection and all of their fields. In fact, your getters and setters can work with any object and your Visualforce page automatically has access to all of the instance attributes and action methods on that object.

Getters and setters seem simple, but the previous example really seems like a lot of code to just allow our Visualforce page to work with a single object. This is why Apex also includes a shortcut for getters and setters called properties. A property combines the instance of our variable with the getter and setter methods and greatly reduces the amount of code you need to write. Take a look at the property equivalent to the previous code block:

//Declare your property and you're done
Account myAccount {get;set;}

Yes, we are serious that one line of code replaces the variable and the getter and setter methods. It really doesn't get more efficient than that! Now, if you've been paying attention, you will see a flaw in our property. If our myAccount variable is null, our Visualforce page can get a nasty error. Our getter took care of that possibility, but our property does not. There are two options to get around this. Let's look at the first option:

//Declare your property with logic and you're done
Account myAccount {
get{
if ( myAccount == null ){
myAccount = new Account();
}
}
set;
}

In this property, we expanded the getter to include our logic. This combines the getter method with the property concept. Although perfectly legitimate, the second option is actually the most common one, and this is to set the value of your attributes in your class' constructor, as shown in the following lines of code:
 

public class myCustomController{
//Here's our property
Account myAccount {get;set;}
//Here's our constructor
public myCustomController(){
myAccount = new Account();
}
}

The key to this code block is in knowing that constructors are called prior to getters. This allows us to execute our code before the Visualforce page communicates with our controller. This pattern of using properties and setting their initial state in the constructor is very common and uses a lot less code. In fact, most classes used with Visualforce have a whole list of properties at the top of the class.


i hope it helps you to understand it properly
thanks
Mark it best answer if it helps you so it make proper solution for others