empty set

Definitions

The unique set that contains no elements, denoted ∅ or {}.

Equivalents

Examples

“Start⟳ with the empty set⟳. We define⟳ it as “the set⟳ of all objects not equal to themselves,” since there are no such objects. All empty sets have⟳ the same members—no members at all! Therefore, as sets they're identical, by definition of identity of sets. In other words, there's only one empty set⟳. This unique empty set⟳ is our building block⟳. Next comes the set⟳ whose only member is—the empty set⟳. This set⟳ is not empty.”

“The existence of the empty set⟳ is entailed, in the presence of the axiom scheme of separation, by the existence of any set⟳ whatever.”

“A topological space may be very sparsely endowed with open⟳ sets. As we know⟳, some spaces have⟳ only two, the empty set⟳ and the full space.”

“Their existence can be generated on the basis of the empty set⟳, since one is simply the set⟳ of the empty set⟳, two is simply that set⟳ plus the empty set⟳ and so forth.”

CEFR level