A JEE question throws this at you: "Is the set \{\{\,\}\} empty?" Half the class ticks "yes" because it looks empty — all those braces, no numbers. The other half stares at it, counts braces, panics. Both are missing the point. The question is not about emptiness; it is about levels of braces, and reading those levels is a pure recognition skill.

This article is the drill for that skill. Once you see how to unfold nested braces one layer at a time, the "is \{\{\,\}\} empty?" question — and its entire family — becomes a thirty-second sight-read.

The one rule

Every pair of curly braces names exactly one set. Every comma-separated item inside those braces is one element of that set.

That is it. Two sentences. The rule is recursive — if one of the elements is itself a pair of curly braces, that inner pair is a separate set, and it counts as one element of the outer set.

The canonical triple

Keep these three objects in your head, because almost every exam question is a remix of them.

All three look similar, but their cardinalities are 0, 1, 1 — and \varnothing is not equal to \{\varnothing\}, despite the visual similarity.

Three nested empty-set objects with boxes showing cardinalityThree rectangles side by side, each representing a set. The first is a rectangle labelled the empty set with nothing inside and cardinality zero. The second is a rectangle containing a smaller rectangle labelled empty set, with the outer rectangle labelled cardinality one. The third is a rectangle containing a rectangle containing a rectangle, with the outer rectangle labelled cardinality one. The cardinality labels are prominent below each figure. (nothing inside) ∅ or { } |A| = 0 empty set {∅} |A| = 1 empty {{∅}} |A| = 1
The three near-identical symbols laid out as boxes. The first has no inner box (cardinality $0$). The second has exactly one inner box (cardinality $1$), even though that inner box is empty. The third has one inner box too (cardinality $1$), but that inner box itself contains another empty box.

The counting drill

When a question shows you a set that looks like nested braces, run this drill:

  1. Find the outermost pair of curly braces. That is the set the question is asking about.
  2. Count the comma-separated items inside those outermost braces. Each item — whether a number, a letter, or an inner set — counts as one element.
  3. Do not descend into inner sets unless the question asks. What is inside \{\varnothing\}'s inner set does not affect \{\varnothing\}'s cardinality; \{\varnothing\} has one element, and that element happens to be the empty set.

Walked examples

Example A. What is |\{\varnothing\}|?

Outermost braces enclose one item (the symbol \varnothing). So |\{\varnothing\}| = 1.

Example B. What is |\{1, \{1\}\}|?

Outermost braces enclose two items: the number 1, and the set \{1\}. Two comma-separated items, so cardinality 2. Note: the two items are different objects — the number 1 is not the same as the set \{1\}, and they each count once.

Example C. What is |\{\varnothing, \{\varnothing\}\}|?

Outermost braces enclose two items: \varnothing and \{\varnothing\}. Cardinality 2.

Example D. Is the set \{\{\,\}\} empty?

The outermost braces enclose one item — the pair of inner empty braces, which is itself the empty set. So \{\{\,\}\} = \{\varnothing\}, which has one element. Not empty.

Why the trap works: the inner \{\,\} is empty, so a reader skimming the expression sees "empty... so the whole thing is empty." But the empty inner set is still something, and the outer braces wrap that something. Emptiness does not propagate outward.

Example E. What is |\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}|?

Outermost braces enclose three items, separated by two top-level commas: \varnothing, \{\varnothing\}, and \{\varnothing, \{\varnothing\}\}. The last item is a nested set with two elements, but from the outer perspective it is one thing. Cardinality 3.

(Incidentally, this is the von Neumann construction of the number 3 in set-theoretic foundations — 0 = \varnothing, 1 = \{\varnothing\}, 2 = \{\varnothing, \{\varnothing\}\}, 3 = \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}. The cardinality of the set that represents n is exactly n, by design.)

Membership inside nested sets

Related family of questions: given a nested set, is some candidate an element?

Three questions, three near-identical shapes, answers \in yes / no / yes and a subset question thrown in. The only way to get these right consistently is to read the outermost braces first and check the listed items, without descending unprompted.

The misconception in two lines

The single sentence that causes most of these errors: "Empty braces inside mean the whole set is empty." Wrong. Empty braces as one of the listed items just mean that one of the items is the empty set — the set still contains that item.

A quicker way to see it: ask what is between the outermost pair of braces. If that region has any content at all — even a single pair of inner braces — the outer set is not empty.

The exam reflex

When a JEE or NEET multiple-choice question shows you something like A = \{\varnothing\} with options "(a) A is empty, (b) |A| = 1, (c) |A| = 2, (d) A = \varnothing" — your fingers should go to option (b) without hesitation. \{\varnothing\} is a one-element set; its one element is \varnothing. Options (a) and (d) are the classic trap, designed for students who confuse "inner emptiness" with "outer emptiness."

Let $A = \{\varnothing, \{1\}, \{1, 2\}, \{\varnothing\}\}$. Find $|A|$.

Outermost braces enclose four comma-separated items: \varnothing, \{1\}, \{1, 2\}, and \{\varnothing\}. Each counts as one element of A, regardless of what is inside.

Result. |A| = 4.

The inner sets' own cardinalities (0, 1, 2, 1) are irrelevant to the outer count. All that matters is how many comma-separated items sit inside the outermost braces.

Related

Sets — Introduction · Is the Power Set of the Empty Set Also Empty? · Can a Set Contain Another Set as an Element? · What's the Difference Between {0} and ∅?