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In Mathematics and more specifically Set Theory , the empty set is the unique Set which contains no elements. In Axiomatic Set Theory it is postulated to exist by the Axiom Of Empty Set . The empty set is also sometimes called the '''null set''', but because Null Set means something else in Measure Theory , that term is generally avoided in current work.

Various possible properties of sets are Trivially true for the empty set.


NOTATION


The empty set is denoted by either one of the symbols " arnothing" or "\emptyset", derived from the letter Ø in the Danish And Norwegian Alphabet , introduced by the Bourbaki Group (specifically André Weil ) in 1939. Earliest Uses of Symbols of Set Theory and Logic Another common notation for the empty set is "{}".


PROPERTIES


(Here we use Mathematical Symbol s.)

  • For Any set ''A'', the empty set is a Subset of ''A'':

  • : ∀''A'': ∅ ⊆ ''A''

  • For any set ''A'', the Union of ''A'' with the empty set is ''A'':

  • : ∀''A'': ''A'' ∪ ∅ = ''A''

  • For any set ''A'', the Intersection of ''A'' with the empty set is the empty set:

  • : ∀''A'': ''A'' ∩ ∅ = ∅

  • For any set ''A'', the Cartesian Product of ''A'' and the empty set is empty:

  • : ∀''A'': ''A'' × ∅ = ∅

  • The only subset of the empty set is the empty set itself:

  • : ∀''A'': ''A'' ⊆ ∅ ⇒ ''A'' = ∅

  • The number of elements of the empty set (that is its Cardinality ) is Zero ; in particular, the empty set is Finite :

  Any Axiom That States The Existence Of Any Set Will Imply The Axiom Of Empty Set, Using The "http://wwwinformationdelightinfo/information/entry/axiom_schema_of_separation" class="copylinks">Axiom Schema Of Separation For example, if ''A'' is a set then the axiom schema of separation allows the construction of the set ''B'' = {''x'' in ''A'' ''x'' ≠ ''x''}, which can be defined to be the empty set