The Empty Set: A Unique Subset of All Non-Empty Sets

The concept of the empty set is a foundational principle in set theory. First, let's define the empty set: it is the unique set with no elements. It can be denoted as:

What is the Empty Set?

The empty set, represented by the symbol ? or { }, is the unique set with no elements. Mathematically, it can be defined as:

?  {x | x ≠ x}

This statement is vacuously true because there are no elements in the empty set to form a counterexample.

The Axiom of the Empty Set

The existence of this unique set is posited by the Axiom of the Empty Set and is formulated as follows:

exists; ; x ; forall ; y : ;  lnot (y in ; x)

This means there exists an x such that for all y, y is not an element of x. In simpler terms, there exists a set such that no element is a member of it.

Subset of Any Set

An important property of the empty set is that it is a subset of any set S. This is because given a set A, every element of the empty set is also an element of A. This is a vacuously true statement, as there are no elements in the empty set to form a counterexample.

Power Set and the Empty Set

The power set P(A) of a set A is the collection of all subsets of A. A well-known theorem states that the cardinality (number of elements) of the power set of a finite set A, denoted ( n_{P(A)} 2^{n_A} ), is 2 times the cardinality of A. When A is the empty set, ( n_A 0 ), thus ( n_{P(A)} 2^0 1 ). This means there is exactly one subset of the empty set, which is the empty set itself.

Subsets and Proper Subsets

In the standard definition, B is a subset of A if all the members of B are members of A. Therefore, every set is a subset of itself. Consequently, the empty set (φ) is a subset of itself. While this may seem odd, it follows from the definitions. By thinking intuitively, since the empty set has no members, it is true that every member of the empty set is a member of itself. In a similar sense, every member of the empty set is a member of every set, making the empty set a subset of every set.

Subsets could be defined differently, but the standard definition is more useful, as it simplifies theorems by not requiring exceptions. However, there is a concept of 'proper subset', where B is a proper subset of A if it is a subset of A but not equal to A. This can be defined more concretely as: every member of B is a member of A, but there is at least one member of A that is not a member of B. The empty set is not a proper subset of itself but is a proper subset of every other set.