Empty set symbol (∅)

How to read it

OpenStax’s Contemporary Mathematics describes the empty set as written either with a pair of braces, { }, or with a zero with a slash through it, ∅, and uses the two interchangeably. Its size is zero. Oscar Levin’s Discrete Mathematics writes ∅ = { }, the set containing no elements. Something that looks empty may not be: {0} has one element, the number 0, so {0} ≠ { } (OpenStax). In the same way, Richard Hammack’s Book of Proof warns against writing {∅} for ∅: one is an empty box, the other a box with an empty box inside it. Unicode names the character EMPTY SET (U+2205), also called null set, and notes that linguists use it for a null morpheme, a “zero”.

Also searched as: {}, ∅, empty set, empty set vs phi, null set symbol, zero with a slash.

∅, Ø or φ?

Three different characters look alike. The empty set ∅ (U+2205) is a mathematical sign. The letter Ø (U+00D8, small ø) is the Danish, Norwegian and Faroese “o slash” it was modelled on; Unicode cross-refers the two. The Greek letter phi φ is a third: Unicode notes that in mathematics the loopy φ is preferred, to contrast with the stroked ϕ (U+03D5), and some fonts make that stroked phi look much like ∅. A fourth, ⌀ DIAMETER SIGN (U+2300), marks a diameter on technical drawings; Unicode cross-refers it to both ø and ∅.

Where you’ll see it

As the answer when two sets share nothing: the Intersection symbol (∩) of disjoint sets is ∅ (OpenStax). And inside other sets: OpenStax notes that the empty set is a proper Subset symbols (⊆, ⊂) of every set except itself.