Notation
A symbol for empty set
Common notations for the empty set include "{}," "" and "" The latter two symbols were introduced by the Bourbaki group (specifically Andre Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabet. Other notations for the empty set include "Λ", "0", and "‣"
Properties
By the principle of extensionality, two sets are equal if they have the same elements; therefore there can be only one set with no elements. Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set."
The mathematical symbols employed below are explained here.
For any set A:
The empty set is a subset of A:
∀A: ∅ ⊆ A
The union of A with the empty set is A:
∀A: A ∪ ∅ = A
The intersection of A with the empty set is the empty set:
∀A: A ∩ ∅ = ∅
The Cartesian product of A and the empty set is empty:
∀A: A × ∅ = ∅
The empty set has the following properties:
Its only subset is the empty set itself:
∀A: A ⊆ ∅ ⇒ A = ∅
The power set of the empty set is a set containing only the empty set:
2∅ = {∅}
Its number of elements (that is, its cardinality) is zero. Moreover, the empty set is finite:
∅ = 0
The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, we use sets to model the natural numbers. In this context, zero is modelled by the empty set.
For any property:
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