The symbol \varnothing shows up in your first chapter on sets and quietly insists on being taken seriously. A set is supposed to be a collection of objects, and now you are told there is a set with no objects in it. That sounds like writing "a queue with no people" and then asking which person is at the front. If nothing is in it, in what sense is the set even there?
This is a great question to pick at, because the answer forces you to upgrade your mental picture of what a set is. You will come out of it understanding sets more deeply than most students who breezed past the definition.
The short reply
The empty set \varnothing is the set whose membership rule is "nothing belongs." It has cardinality 0. It is a completely legitimate set — in fact, it is the unique set with zero elements, and modern set theory takes its existence as one of its opening axioms. Without it, half the operations you care about (intersection, solution sets, conditional probability) stop being well-defined.
What a set actually is
The confusion comes from translating "set" as "a bag of stuff." If the bag is empty, there is no bag — just air. And "air" doesn't sound like a mathematical object.
But a set is not defined by what it contains; it is defined by its membership rule. Hand me any object x in the universe — the number 3, your phone, the poet Kabir, the number \pi. The set A either contains x or doesn't. That yes-or-no verdict, for every possible x, is everything A is.
The empty set corresponds to the rule that says no to every candidate. For every x you can name, x \notin \varnothing. That is a perfectly consistent rule — arguably the simplest rule there is, because stating it takes no information about the universe.
Why: think of a set as a yes/no answering machine. The answering machine for \{2, 4, 6\} says "yes" to 2, "yes" to 4, "yes" to 6, "no" to everything else. The answering machine for \varnothing is the one that answers "no" to every query. It is still a well-defined machine.
Why outlawing it breaks mathematics
Suppose you dug your heels in and insisted every set must have at least one element. Watch what collapses.
Take two perfectly ordinary sets of integers:
What is A \cap B? They share no element. So the intersection would have to be "not a set." But intersection is supposed to combine two sets and return a set, always. If you outlaw \varnothing, every theorem and proof that uses intersection would have to carry a footnote: "provided the two sets have at least one element in common." Every JEE probability question where two events turn out to be mutually exclusive would hit the same snag.
The cleanest fix is to let \varnothing exist and let intersection always return a set. That single move keeps the whole machinery of set operations uniform.
A visual that actually helps
Imagine filtering. You have a universe of numbers U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, and you apply a filter: "keep only the multiples of 11."
The filter is a mechanical process. Numbers go in, matching numbers come out. When zero numbers match, the output bin is still a bin — it is just empty. That empty bin is \varnothing. You would not say "the bin didn't happen." The bin happened; nothing landed in it.
Every time you write \{x \in \mathbb{N} \mid x^2 < 0\} or "the set of real solutions to x^2 + 1 = 0," you are running a filter that catches nothing. The result is \varnothing, full stop.
Why there is only one empty set
A subtle but pretty fact: there isn't a "small empty set" and a "big empty set." There is exactly one.
Two sets are equal when they have the same elements. If A and B are both empty, then for every object x, the statement "x \in A" is false and "x \in B" is false. They agree on every single question. So A = B. There is only one empty set — and this is why mathematicians write \varnothing with its own symbol, as if it were a proper noun.
Why: "same elements" is the only way sets differ. If two sets have no elements, they have the same elements trivially — there are no elements to disagree about.
"But it doesn't feel like a collection"
The honest answer is: you're right, in everyday English, "collection of nothing" is awkward. Mathematics deliberately stretches the word "set" past its ordinary meaning to make the theory uniform. It is like how 0 stretched the meaning of "number" past the counting numbers, or -3 stretched it past positive quantities. Each extension felt weird when it was introduced and now feels obvious.
The empty set is the analog of 0 in the world of sets — the thing that makes the whole algebra work smoothly and that every operation can land on cleanly. Just as you wouldn't now say "0 isn't really a number because it doesn't count anything," you learn to not say "\varnothing isn't really a set because it doesn't collect anything."
JEE-level payoff
In probability, P(A \cap B) = 0 often means A \cap B = \varnothing — mutually exclusive events. In algebra, "the solution set of this equation is \varnothing" is the crisp way of saying "there is no solution." In coordinate geometry, "the intersection of these two lines is \varnothing" says they are parallel.
Being comfortable with \varnothing isn't a pedantic nicety. It is the reason your formulas keep applying even at the edge cases where nothing qualifies — and edge cases are where exam questions love to live.
Related: Sets — Introduction · Is the Power Set of the Empty Set Also Empty? · Does Every Set Have to Contain at Least One Element? · Difference Between {0} and the Empty Set