My question is short and concise. Here it goes -
In my book the definition of a set is given as a well defined collection of things and in mathematicse they are well defined collection of mathematical objects. Then why is $\emptyset$ which has nothing is even considered as a set. Is it merely a mathematica convention or is it that it has a special significance ?
Though it is pretty general, I want to know the reason behind it. Thanks for your help .
The existence of the empty set is one of the Zermelo-Frankel axioms of set theory.
One can argue whether or not the concept of the "empty set" violates one's intuition. I can give you an intuitive argument to say that it is not a violation, along the following lines: "Think of a set as the contents of a bag. Just because the bag has no contents doesn't mean it's not a bag."
But that's not the real point, because even if the "empty set" does violate intuition, there is still a good reason to include it in our mathematical language (it often happens that when a mathematical concept is formalized, some of the axioms/rules/concepts that are needed in order for the formalization to work are not as intuitive as one might want; think of the law of logic that says $P \implies Q$ is true whenever the premise $P$ is false and the conclusion $Q$ is true).
What's the reason? Among other possible reasons, one can say that similar to how the theory of addition is simpler when one introduces zero, the theory of sets is simpler when one introduces the emptyset. One wants to be able to define the binary operation of intersection $A \cap B$ for all pairs of sets $A$ and $B$. The definition is: $$A \cap B = \{x \bigm| \text{$x \in A$ and $x \in B$}\} $$ However, what if there does not exist any $x$ such that $x \in A$ and $x \in B$? In that case, the only candidate for $A \cap B$ is the empty set, so if the empty set does not exist then $A \cap B$ is not always defined.
The other answers tend to treat this from the perspective of ZFC, a specific way of formalizing the ideas behind set theory; below I give a more informal approach, which has the benefit of applying to set theories other than ZFC.
they are well-defined collections of mathematical objects
The emptyset is a well-defined collection of mathematical objects - it's certainly well-defined, and every thing in the emptyset is a mathematical object! (Every thing in the emptyset is also a purple unicorn, but that's fine - there's no rule saying "a set has to have something which isn't a purple unicorn.")
Ultimately, this comes down to the question, "What is a collection?" This ultimately comes down to relevance: for any mathematical object, there are questions about it which simply don't make sense. E.g. is $17$ blue? Informally, a statement like "A set is a collection" tells us what a set is by telling us what it does: the only meaningful questions you can ask of sets are those which revolve around membership ("Is $x$ in $A$?" "Does $A$ have at least three elements?" etc.). This is also the motivation behind the Axiom of Extensionality, which states that two sets with the same elements are the same.
Now it should be clear that there's no problem with a set being empty: that just means that whenever I ask "Is $x$ in this set?," the answer I get back is "no."
Now it takes some work to make everything I've said above precise, but the intuitive meaning of it should be clear, and hopefully it's somewhat persuasive. Personally I would say that this kind of "limiting meaningfulness" approach is actually of fundamental philosophical importance to mathematics, and there is a philosophical debate around this point, but that's going far afield.
If the empty set did not exist, then we couldn't perform subset selection without lots of tedious extra writing. We couldn't say that $A = \{x \in B: \varphi(x) \}$ exists (where $\varphi$ is some property), without first proving that there is indeed some $x \in B$ such that $\varphi(x)$.
In "usual" set theory, the existence of $\emptyset$ is assumed by a specific axiom.
Alternatively, we may assume that there is at least one set $a$; then, the correct way to prove that a "well defined collection" exists [by way of Axiom of Separation ] can be applied to $a$ proving that its subset $\{ x \mid x \in a \text { and } x≠x \}$ exists.
Having proved that an empty set exists, we may apply Extensionality to prove that it is unique, and thus we may introduce a "name" for it, i.e. for the empty set: $\emptyset$.
For a more "intuitive" example, consider the set $\mathbb N$ of natural numbers and consider its subset of all and only those numbers $n$ such that $n < 0$.
It is a "well defined collection" that has no numbers (no elements) inside.
Thus, we have defined a "reasonable" empty subset of $\mathbb N$.