Subset symbols (⊆, ⊂)
⊆ means “is a subset of”: A ⊆ B says that every element of A is also in B. ⊂ usually means “is a proper subset of”: A is inside B and B has at least one element A doesn’t.
Easily confused with
How to read it
OpenStax’s Contemporary Mathematics defines A ⊆ B as every member of A also being a member of B, so every set is a subset of itself, B ⊆ B. It keeps ⊂ for a proper subset, where B has other members not in A: a knife and fork taken from the cutlery drawer are a proper subset of the drawer. Oscar Levin’s Discrete Mathematics uses the two signs the same way, defining A ⊂ B as A ⊆ B but A ≠ B. A ⊆ B is a statement, true or false, where A ∩ B is a set (Levin). The empty set is a proper subset of every set except itself (OpenStax).
Also searched as: ⊂, ⊃, ⊆, ⊇, proper subset symbol, subset or equal to, subset symbol, superset symbol.
Which sign means which
Books agree on ⊆ but not always on ⊂. OpenStax and Levin read ⊂ as a proper subset. NIST’s Digital Library of Mathematical Functions glosses ⊂ as “is contained in” and ⊆ as “is, or is contained in”. Unicode names ⊂ simply SUBSET OF, “included in set”, and has a third character, ⊊ SUBSET OF WITH NOT EQUAL TO (U+228A), that says “proper” unmistakably. Where it matters, the text defines the sign it uses.
Turned the other way
⊇ and ⊃ are the same relations read from the other side: B ⊇ A, “B is a superset of A” (Unicode, SUPERSET OF, “includes in set”). The same ⊃ is also the horseshoe, an older sign in logic for “if … then” (forall x: Calgary).
Read next
- Next in Set theory symbolsSet-builder notation ({x | …})Set-builder notation describes a set by a rule: {x | x > 2} is read “the set of all x such that x is greater than 2”. The vertical bar (or a colon) means “such that”.
- The same shape in Asian flagsFlag of PakistanPakistan’s flag is dark green with a white vertical band at the pole and a white crescent and five-pointed star on the green. Pakistan’s Consulate General in Jeddah gives the meanings: green for Islam and the Muslim majority, white for the religious minorities, the crescent for progress and the star for enlightenment.
More set theory symbols
The same shape elsewhere
Browse symbols drawn with curves and waves or lines.
Sources
- Contemporary Mathematics, 1.2 Subsets, OpenStax (Rice University), Subsets and proper subsets
- Discrete Mathematics, 0.3 Sets, Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin, University of Northern Colorado (Open Math Books), Set theory notation
- NIST Digital Library of Mathematical Functions: Introduction (Common Notations and Definitions), U.S. National Institute of Standards and Technology, Common Notations and Definitions
- 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, Set inclusion
- Mathematical Operators: Range 2200–22FF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium, Relations, 2282–228B
- forall x: Calgary. An Introduction to Formal Logic, P. D. Magnus, Tim Button, Robert Trueman and Richard Zach, forall x: Calgary (Fall 2025 edition), Open Logic Project, University of Calgary, Appendix A.2 Alternative symbols
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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