At a glance, \{0\} and \varnothing look like twins of nothingness. One contains the number zero; the other contains no numbers. Zero is "nothing" in arithmetic. So aren't both of these sets "empty-ish," in some sense? And doesn't that make them equal?
They are not equal, and the reason is worth getting right on day one. The difference between \{0\} and \varnothing is the tiny fence that separates the thing from the container holding the thing. Blur it and a lot of later problems — power sets, probability, the cardinality of intersections — will make no sense.
The one-sentence reply
\{0\} is a set with one element, namely the number 0. The empty set \varnothing is a set with zero elements. They differ in their cardinality: |\{0\}| = 1, while |\varnothing| = 0.
That is the whole distinction. The rest of this article is spent making it feel obvious.
Container and contents are different things
The trap is that the word "zero" is doing two different jobs in English.
- "Zero" can name the number 0 — an object that sits on the number line between -1 and +1.
- "Zero" can also mean nothing, as in "zero apples in the basket."
When you write \{0\}, you are using the first sense: you are building a set whose only element is that specific number, the one on the number line. When you write \varnothing, you are using the second sense: no elements at all.
A shopping-basket picture makes this crisp:
The basket on the left has one token inside — the token happens to have "0" written on it, but a token is still a token. The basket on the right is empty. The baskets are not the same. The number of items in them is different.
Why: think of the outer \{\ldots\} braces as the basket and the contents between them as tokens. In \{0\} there is one token (a zero). In \varnothing there are no tokens. Counting tokens, you get 1 and 0 respectively.
A cleaner test: membership
Check which objects are elements of each set.
- Is 0 \in \{0\}? Yes. The set \{0\} lists 0 as its single element.
- Is 0 \in \varnothing? No. The empty set has no elements — not 0, not anything.
If \{0\} and \varnothing were equal, they would have the same elements. Since 0 is in one and not the other, they differ. End of argument.
Cardinality, explicitly
Cardinality |S| counts the elements of S.
Notice the ladder: the empty set has 0 elements; \{0\} adds one token (the number zero) and jumps to 1; \{0, 1\} adds another and jumps to 2. The name of the token doesn't shrink the count. A token labelled "0" is as much an element as a token labelled "1" or "\pi."
This is also why \{\varnothing\} — a set whose one element is itself the empty set — has cardinality 1, not 0. The container has one object inside (the empty set), which is different from the container being empty.
Why this matters later
Power sets. The power set \mathcal{P}(\varnothing) = \{\varnothing\}. That is, the power set of the empty set is not empty — it is a one-element set whose single element is \varnothing. If you confused \{\varnothing\} with \varnothing, you would claim |\mathcal{P}(\varnothing)| = 0, violating the universal formula |\mathcal{P}(A)| = 2^{|A|}, which gives 2^0 = 1 here. See Power Set of the Empty Set.
Probability. If A is the event "the die shows a zero on a standard 1{-}6 die," then A = \varnothing, and P(A) = 0. If A is the event "the die shows zero defects" on a special die that includes 0, then A = \{0\}, and P(A) = 1/6. Confusing the two confuses a zero-probability event with a specific outcome event.
Solution sets. The equation x^2 = 0 has solution set \{0\} (one solution, namely zero). The equation x^2 = -1 in \mathbb{R} has solution set \varnothing (no real solutions). Writing one for the other changes the meaning entirely.
A tempting wrong move
Some students argue: "zero is nothing, so the set containing zero is the set containing nothing, which is empty." This conflates zero as a number with zero as a count. Zero on the number line is a full-fledged mathematical object — you can add it, multiply by it, use it as the y-intercept of a line. It is as real an element as 5 or \pi. A set with zero in it is a set with one element, not no elements.
The parallel in English is: a box containing one empty envelope is not an empty box. The envelope is an object; it is a bit weird that the envelope is empty, but the box is still holding something — one envelope.
Quick drill
Classify each set's cardinality.
- \varnothing — cardinality 0.
- \{\varnothing\} — cardinality 1. (One element: \varnothing.)
- \{0\} — cardinality 1. (One element: 0.)
- \{0, \varnothing\} — cardinality 2. (Elements: 0 and \varnothing.)
- \{\{0\}\} — cardinality 1. (One element: \{0\}.)
- \{0, \{0\}\} — cardinality 2. (Elements: 0 and \{0\}.)
The trick each time is to count what is listed between the outermost braces. Nothing inside an element counts toward the outer cardinality. For more on this, see Cardinality Meter and Can a Set Contain Another Set?.
The mindset
Two habits will save you dozens of mistakes later:
- Read the outer braces as "the basket"; read the contents as "the tokens."
- When in doubt, compute cardinality — the count of tokens tells you whether two sets are even in the same league.
With those two reflexes, \{0\} and \varnothing stop looking similar and start looking as different as a basket with one apple and an empty basket.
Related: Sets — Introduction · The Empty Set Isn't Really a Set — How Can Nothing Be a Set? · Is the Power Set of the Empty Set Also Empty? · Cardinality Meter