Empty set
The unique set with no elements, fundamental to mathematics.
The empty set, also called the void set, is the unique set in mathematics that contains no elements. Its cardinality, or size, is zero. The empty set is fundamental to set theory and appears across many areas of mathematics, often serving as the basis for defining natural numbers and as an initial object in category theory.
- definition
- The unique set with no elements
- cardinality
- Zero
- common_notations
- { }, ∅, ∅
- introduced_by
- Bourbaki group (specifically André Weil) in 1939
- symbol_inspired_by
- The letter Ø in Danish and Norwegian alphabets
- also_called
- Null set (in some textbooks, though distinct in measure theory)
Lore & Background
The empty set is the only set with no elements, and by the principle of extensionality, there can be only one such set. Its power set contains only the empty set itself. For any set A, the empty set is a subset of A; the union of A with the empty set is A; the intersection of A with the empty set is the empty set; and the Cartesian product of A and the empty set is the empty set. Many properties hold vacuously for the empty set, such as 'for every element of the empty set, property P holds.'
Reader's Guide
The empty set is a cornerstone of modern mathematics. In set theory, it is used to model the number zero in the von Neumann construction of ordinals. In topology, the empty set is both open and closed in any topological space, and it is compact. In category theory, the empty set is the unique initial object in the category of sets. Its supremum in the extended real numbers is negative infinity, and its infimum is positive infinity. The empty set also gives rise to the empty function and the empty product, which is defined as one. Its notation, introduced by the Bourbaki group, is widely used across mathematical disciplines.
Did You Know?
- The empty set is the only set whose power set contains only itself.
- The symbol ∅ was introduced by the Bourbaki group in 1939, inspired by the Danish and Norwegian letter Ø.
- In the extended real numbers, the supremum of the empty set is negative infinity and the infimum is positive infinity.
- The empty set can be considered a derangement of itself, as it has exactly one permutation (0! = 1) and no element retains its original position.
More in Sets And Logic 1-24
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