Set theory symbols
The symbols of sets, from ∈ and ⊆ to ∪, ∩ and ∅, the number sets ℕ, ℤ, ℚ, ℝ and ℂ, and set-builder notation, what each one means and how it is read.
Test yourself: Set theory symbols quiz
Membership and subsets 3 symbols
Operations 4 symbols
Special sets 2 symbols
About set theory symbols
Set theory symbols describe collections of things and how they relate. A set is written in curly braces, either by listing its elements, {1, 2, 3}, or by giving a rule in set-builder notation. ∈ says that something is a member of a set, and ⊆ and ⊂ that one set sits inside another. ∪, ∩, the set difference ∖ and the complement A′ make new sets out of old ones, and ∅ is the set with nothing in it. The double-struck letters ℕ, ℤ, ℚ, ℝ and ℂ name the standard sets of numbers. Each page says what one symbol means, how it is read, and where books disagree, as they do over ⊂ and over whether 0 is a natural number. The readings are from OpenStax’s Contemporary Mathematics, Oscar Levin’s Discrete Mathematics: An Open Introduction, NIST’s Digital Library of Mathematical Functions and the Unicode charts; the histories are from Jeff Miller’s Earliest Uses of Various Mathematical Symbols, published by the University of St Andrews.
Sets and statements
Levin draws a line between two kinds of symbol. The operations (union, intersection, difference and complement) produce sets. ∈ and ⊆ make statements, which are true or false. A ∩ B is a set, while A ⊆ B is true or false, in the same way that 3 + 2 is a number and 3 ≤ 2 is false.
Where the symbols came from
Set theory came into being in the late nineteenth and early twentieth centuries, largely the creation of Georg Cantor, and most of its basic symbols were introduced between 1880 and 1920 (Miller). Giuseppe Peano wrote ∩ and ∪ in 1888 and an epsilon for membership, the ancestor of ∈, in 1889. The empty set sign came from the Bourbaki group in 1939, modelled on a Norwegian letter. Sets and logic grew up together: OpenStax calls union a logical inclusive OR and intersection a logical AND, the set versions of ∨ and ∧ in Logic symbols.
Sources
- Discrete Mathematics, 0.3 Sets, Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin, University of Northern Colorado (Open Math Books), Set operations
- Earliest Uses of Symbols of Set Theory and Logic, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, Introduction; Intersection and union; Membership; Null set
- Contemporary Mathematics, 1.1 Basic Set Concepts, OpenStax (Rice University), Basic set concepts
- Contemporary Mathematics, 1.4 Set Operations with Two Sets, OpenStax (Rice University), Intersection; union
- NIST Digital Library of Mathematical Functions: Introduction (Common Notations and Definitions), U.S. National Institute of Standards and Technology, Common Notations and Definitions
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research