The Empty Set

Axiom: Existence of an Empty Set

There exists a set with no elements.

$$\exists A\forall x(x\notin A)$$

NOTATION

$$\varnothing \varnothing \{\}$$

Theorem: Uniqueness of the Empty Set

There is only one empty set.

PROOF

Let $A$ and $B$ be two empty sets. They have no elements and so the statement

$$x\in A⟺x\in B$$

is vacuously true. Therefore, $A=B$.

Theorem

The empty set is a subset of all sets.

PROOF

Let $A$ be a set.

Suppose that the empty set $\varnothing$ is not a subset of $A$. Then there must exist an element $e\in \varnothing$ which is not an element of $A$, i.e. $e\notin A$. This is a contradiction, since the empty set contains no elements.