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Glyph for U+228E
Source: Noto Sans Math

U+228E Multiset Union

U+228E was added in Unicode version 1.1 in 1993. It belongs to the block U+2200 to U+22FF Mathematical Operators in the U+0000 to U+FFFF Basic Multilingual Plane.

This character is a Math Symbol and is commonly used, that is, in no specific script. The character is also known as z notation bag addition.

The glyph is not a composition. It has no designated width in East Asian texts. In bidirectional text it acts as Other Neutral. When changing direction it is not mirrored. The word that U+228E forms with similar adjacent characters prevents a line break inside it. The glyph can be confused with one other glyph.

This character was planned to be used for the U+1234 notation of Unicode code points. But due to the chicken-egg-problem of slow adoption of the then-young Unicode it was decided to use the “ASCIIfied” version U+ instead.

The Wikipedia has the following information about this codepoint:

In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements a and b, but vary in the multiplicities of their elements:

  • The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset.
  • In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.
  • In the multiset {a, a, a, b, b, b}, a and b both have multiplicity 3.

These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, the order in which elements are listed does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted by [a, a, b].

The cardinality of a multiset is the sum of the multiplicities of all its elements. For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.

Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth.: 694  However, the concept of multisets predates the coinage of the word multiset by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite.: 694 


System Representation
UTF-8 E2 8A 8E
UTF-16 22 8E
UTF-32 00 00 22 8E
URL-Quoted %E2%8A%8E
HTML hex reference ⊎
Wrong windows-1252 Mojibake ⊎
HTML named entity ⊎
HTML named entity ⊎
alias z notation bag addition
LATEX \uplus

Related Characters



Complete Record

Property Value
Age (age) 1.1 (1993)
Unicode Name (na) MULTISET UNION
Unicode 1 Name (na1)
Block (blk) Mathematical Operators
General Category (gc) Math Symbol
Script (sc) Common
Bidirectional Category (bc) Other Neutral
Combining Class (ccc) Not Reordered
Decomposition Type (dt) none
Decomposition Mapping (dm) Glyph for U+228E Multiset Union
Lowercase (Lower)
Simple Lowercase Mapping (slc) Glyph for U+228E Multiset Union
Lowercase Mapping (lc) Glyph for U+228E Multiset Union
Uppercase (Upper)
Simple Uppercase Mapping (suc) Glyph for U+228E Multiset Union
Uppercase Mapping (uc) Glyph for U+228E Multiset Union
Simple Titlecase Mapping (stc) Glyph for U+228E Multiset Union
Titlecase Mapping (tc) Glyph for U+228E Multiset Union
Case Folding (cf) Glyph for U+228E Multiset Union
ASCII Hex Digit (AHex)
Alphabetic (Alpha)
Bidi Control (Bidi_C)
Bidi Mirrored (Bidi_M)
Composition Exclusion (CE)
Case Ignorable (CI)
Changes When Casefolded (CWCF)
Changes When Casemapped (CWCM)
Changes When NFKC Casefolded (CWKCF)
Changes When Lowercased (CWL)
Changes When Titlecased (CWT)
Changes When Uppercased (CWU)
Cased (Cased)
Full Composition Exclusion (Comp_Ex)
Default Ignorable Code Point (DI)
Dash (Dash)
Deprecated (Dep)
Diacritic (Dia)
Emoji Modifier Base (EBase)
Emoji Component (EComp)
Emoji Modifier (EMod)
Emoji Presentation (EPres)
Emoji (Emoji)
Extender (Ext)
Extended Pictographic (ExtPict)
FC NFKC Closure (FC_NFKC) Glyph for U+228E Multiset Union
Grapheme Cluster Break (GCB) Any
Grapheme Base (Gr_Base)
Grapheme Extend (Gr_Ext)
Grapheme Link (Gr_Link)
Hex Digit (Hex)
Hyphen (Hyphen)
ID Continue (IDC)
ID Start (IDS)
IDS Binary Operator (IDSB)
IDS Trinary Operator and (IDST)
ID_Compat_Math_Continue (ID_Compat_Math_Continue) 0
ID_Compat_Math_Start (ID_Compat_Math_Start) 0
Ideographic (Ideo)
InCB (InCB) None
Indic Mantra Category (InMC)
Indic Positional Category (InPC) NA
Indic Syllabic Category (InSC) Other
Jamo Short Name (JSN)
Join Control (Join_C)
Logical Order Exception (LOE)
Math (Math)
Noncharacter Code Point (NChar)
NFC Quick Check (NFC_QC) Yes
NFD Quick Check (NFD_QC) Yes
NFKC Casefold (NFKC_CF) Glyph for U+228E Multiset Union
NFKC Quick Check (NFKC_QC) Yes
NFKC_SCF (NFKC_SCF) Glyph for U+228E Multiset Union
NFKD Quick Check (NFKD_QC) Yes
Other Alphabetic (OAlpha)
Other Default Ignorable Code Point (ODI)
Other Grapheme Extend (OGr_Ext)
Other ID Continue (OIDC)
Other ID Start (OIDS)
Other Lowercase (OLower)
Other Math (OMath)
Other Uppercase (OUpper)
Prepended Concatenation Mark (PCM)
Pattern Syntax (Pat_Syn)
Pattern White Space (Pat_WS)
Quotation Mark (QMark)
Regional Indicator (RI)
Radical (Radical)
Sentence Break (SB) Other
Soft Dotted (SD)
Sentence Terminal (STerm)
Terminal Punctuation (Term)
Unified Ideograph (UIdeo)
Variation Selector (VS)
Word Break (WB) Other
White Space (WSpace)
XID Continue (XIDC)
XID Start (XIDS)
Expands On NFC (XO_NFC)
Expands On NFD (XO_NFD)
Expands On NFKC (XO_NFKC)
Expands On NFKD (XO_NFKD)
Bidi Paired Bracket (bpb) Glyph for U+228E Multiset Union
Bidi Paired Bracket Type (bpt) None
East Asian Width (ea) neutral
Hangul Syllable Type (hst) Not Applicable
ISO 10646 Comment (isc)
Joining Group (jg) No_Joining_Group
Joining Type (jt) Non Joining
Line Break (lb) Alphabetic
Numeric Type (nt) none
Numeric Value (nv) not a number
Simple Case Folding (scf) Glyph for U+228E Multiset Union
Script Extension (scx)
Vertical Orientation (vo) R