Definition 6 (Empty set). ∅ is the set with no elements at all.
10The empty set
There is a set with no elements whatsoever. We call it the empty set, and we denote it by ∅. Using the list notation from above, we could equally write it {}:
The first thing to prove about such a set usually takes a moment to accept.
Proposition 4 (an empty set is a subset of anything). For any sets X and Y, if X contains no elements, then X ⊂ Y.
If X contains no elements, then every element of X is an element of Y, because all of the zero elements which belong to X satisfy this property.
More generally, a claim of the form “every A is a B”, where there are no As at all, is always counted as true. Mathematicians say that such a claim is vacuously true.
Sometimes a statement is very easy to prove once we already have some other statement in hand. When that happens, we call the easy statement a corollary of the one we are leaning on. The next proposition is a corollary of Propositions 1 (double inclusion) and 4 (an empty set is a subset of anything).
Proposition 5 (the empty set is unique). Any two sets with no elements are identical.
Exercise 2. Prove Proposition 5 (the empty set is unique), as a corollary of Propositions 1 (double inclusion) and 4 (an empty set is a subset of anything).
This is what justifies us in speaking of the empty set rather than an empty set.
The empty set has nothing inside it, so there is nothing for a subset of it to contain either.
Proposition 6 (the only subset of ∅). For any set X, if X ⊂ ∅, then X = ∅.
Suppose for contradiction that there is some x ∈ X. Then since X ⊂ ∅, we have x ∈ ∅, contradicting the definition of ∅. Hence there is no x ∈ X and by Proposition 5 (the empty set is unique) it follows that X = ∅.
One warning while we are here. Do not confuse ∅ with {∅}. The first has no elements at all. The second is a set which contains exactly one element, namely the empty set. So, by the Axiom of Extensionality, they are not the same set.