TITLE: What is an Empty set?
QUESTION [16 upvotes]: We define the term "Set" as,

A set is a collection of objects.

And an "Empty set" as,

An empty set is a set which contains nothing. 

First problem I encountered:
How the definition of "Empty set" is consistent with the definition of "sets" if "Empty set" contains nothing and a "set" is a collection of objects.
Further we discovered in set theory that every set has a subset that is the Null set.
Such as,
If $A=\emptyset$ and $B=\{1,2,3\}$ then,
$$A \subset B$$
Second problem I encountered: 
How and why "No element" is referred and considered as an element as we do in case of null set that is, when we say that every set has a subset that is the Null set?
Third and Last one:
How can a set possess "some thing" and "nothing" simultaneously that is when we say that every set (containing objects) has a subset which is Null set (contains nothing)?

REPLY [3 votes]: People used to argue that $0$ wasn't a number for reasons very much like your arguements about the empty set.
Forget about defining sets. The most important thing about sets is this.
If x is an object and S is a set, then either $x\in S$ is true or $x \notin S$ is true; but never both.
When people say that no element is a member of the empty set they are just, badly, trying to say that $x \in \varnothing$ is always false and $x \notin \varnothing$ is always true.
There is a very basic connection between sets and logic. Learning set theory will help you use and understand logic and vice versa.
It's also very convenient to have an empty set. For example.

If we want to say that sets A and B have no elements in common, then
we can say that their intersection is the empty set.
If we want to say that an equation has no solution, then we can say
that the solution set is empty.

There is no nice way to put this. You are confused because you refuse to change your ideas about the way mathematics works. I am very good at mathematics and I have had that same problem more than once. If you are lucky, someone will explain it so that things in your head click into place. If not, which is the more probable thing, you are going to have to work this out yourself.