# Mathematical Operators

## Miscellaneous mathematical symbols

`2200` ∀ For All
=universal quantifier
`2201` ∁ Complement
`0297` ʗ latin letter stretched c
`2202` ∂ Partial Differential
`2203` ∃ There Exists
=existential quantifier
`2204` ∄ There Does Not Exist
`2203` ∃ `0338` ◌̸
`2205` ∅ Empty Set
=null set
used in linguistics to indicate a null morpheme or phonological "zero"
`00D8` Ø latin capital letter o with stroke
`2300` ⌀ diameter sign
`2206` ∆ Increment
=Laplace operator
=forward difference
=symmetric difference (in set theory)
other symbols may also be used for symmetric difference
`0394` Δ greek capital letter delta
`25B3` △ white up-pointing triangle
`2207` ∇ Nabla
=backward difference
used for Laplacian operator (written with superscript 2)
`25BD` ▽ white down-pointing triangle

## Set membership

`2208` ∈ Element Of
`2209` ∉ Not An Element Of
`2208` ∈ `0338` ◌̸
`220A` ∊ Small Element Of
originates in math pi fonts; not the straight epsilon
`03F5` ϵ greek lunate epsilon symbol
`220B` ∋ Contains As Member
=such that
`220C` ∌ Does Not Contain As Member
`220B` ∋ `0338` ◌̸
`220D` ∍ Small Contains As Member
`03F6` ϶ greek reversed lunate epsilon symbol

## Miscellaneous mathematical symbol

`220E` ∎ End Of Proof
=q.e.d.
`2023` ‣ triangular bullet
`25AE` ▮ black vertical rectangle

## N-ary operators

`220F` ∏ N-Ary Product
=product sign
`03A0` Π greek capital letter pi
`2210` ∐ N-Ary Coproduct
=coproduct sign
`2211` ∑ N-Ary Summation
=summation sign
`03A3` Σ greek capital letter sigma
`2140` ⅀ double-struck n-ary summation

## Operators

`2212` − Minus Sign
`002D` - hyphen-minus
`2796` ➖ heavy minus sign
`2213` ∓ Minus-Or-Plus Sign
`00B1` ± plus-minus sign
`2214` ∔ Dot Plus
`2215` ∕ Division Slash
generic division operator
`002F` / solidus
`2044` ⁄ fraction slash
`27CB` ⟋ mathematical rising diagonal
`2216` ∖ Set Minus
`005C` \ reverse solidus
`27CD` ⟍ mathematical falling diagonal
`29F5` ⧵ reverse solidus operator
`2217` ∗ Asterisk Operator
`002A` * asterisk
`2218` ∘ Ring Operator
=composite function
=APL jot
`00B0` ° degree sign
`0970` ॰ devanagari abbreviation sign
`25E6` ◦ white bullet
`10B39` 𐬹 avestan abbreviation mark
`2219` ∙ Bullet Operator
`00B7` · middle dot
`2022` • bullet
`2024` ․ one dot leader
`221A` √ Square Root
`2713` ✓ check mark
`221B` ∛ Cube Root
`0606` ؆ arabic-indic cube root
`221C` ∜ Fourth Root
`0607` ؇ arabic-indic fourth root
`221D` ∝ Proportional To
`03B1` α greek small letter alpha

## Miscellaneous mathematical symbol

`221E` ∞ Infinity

## Angles

Other angle symbols are found at 299B-29AF.
`221F` ∟ Right Angle
`2220` ∠ Angle
`299F` ⦟ acute angle
`2221` ∡ Measured Angle
`2222` ∢ Spherical Angle
=angle arc

## Relations

`2223` ∣ Divides
=such that
=APL stile
`007C` | vertical line
`01C0` ǀ latin letter dental click
`2224` ∤ Does Not Divide
`2223` ∣ `0338` ◌̸
`2225` ∥ Parallel To
`01C1` ǁ latin letter lateral click
`2016` ‖ double vertical line
`2226` ∦ Not Parallel To
`2225` ∥ `0338` ◌̸

## Logical and set operators

`2227` ∧ Logical And
=wedge, conjunction
`22C0` ⋀ n-ary logical and
`2303` ⌃ up arrowhead
`2228` ∨ Logical Or
=vee, disjunction
`22C1` ⋁ n-ary logical or
`2304` ⌄ down arrowhead
`2229` ∩ Intersection
=cap, hat
`22C2` ⋂ n-ary intersection
`222A` ∪ Union
=cup
`22C3` ⋃ n-ary union

## Integrals

`222B` ∫ Integral
`0283` ʃ latin small letter esh
`222C` ∬ Double Integral
`222B` ∫ `222B` ∫
`222D` ∭ Triple Integral
`2A0C` ⨌ quadruple integral operator
`222B` ∫ `222B` ∫ `222B` ∫
`222E` ∮ Contour Integral
`2A15` ⨕ integral around a point operator
`222F` ∯ Surface Integral
`222E` ∮ `222E` ∮
`2230` ∰ Volume Integral
`222E` ∮ `222E` ∮ `222E` ∮
`2231` ∱ Clockwise Integral
`2232` ∲ Clockwise Contour Integral
`2233` ∳ Anticlockwise Contour Integral
clockwise or anticlockwise arrows do not reverse during layout mirroring

## Miscellaneous mathematical symbols

`2234` ∴ Therefore
`26EC` ⛬ historic site
`2235` ∵ Because

## Relations

`2236` ∶ Ratio
preferred to `003A` : for denotation of division or scale in mathematical use
`003A` : colon
`2237` ∷ Proportion

## Operator

`2238` ∸ Dot Minus
=saturating subtraction
sometimes claimed as notation for symmetric set difference, but `2206` ∆ is preferred

## Relation

`2239` ∹ Excess

## Operator

`223A` ∺ Geometric Proportion

## Relations

`223B` ∻ Homothetic
`2A6B` ⩫ tilde operator with rising dots
`223C` ∼ Tilde Operator
=varies with (proportional to)
=difference between
=similar to
=not
=cycle
=APL tilde
`007E` ~ tilde
`00AC` ¬ not sign
`02DC` ˜ small tilde
`223D` ∽ Reversed Tilde
=lazy S
reversed tilde and lazy S are glyph variants
`223E` ∾ Inverted Lazy S
=most positive

## Miscellaneous mathematical symbol

`223F` ∿ Sine Wave
=alternating current

## Operator

`2240` ≀ Wreath Product

## Relations

`2241` ≁ Not Tilde
`223C` ∼ `0338` ◌̸
`2242` ≂ Minus Tilde
`2243` ≃ Asymptotically Equal To
`2244` ≄ Not Asymptotically Equal To
`2243` ≃ `0338` ◌̸
`2245` ≅ Approximately Equal To
`2246` ≆ Approximately But Not Actually Equal To
`2247` ≇ Neither Approximately Nor Actually Equal To
`2245` ≅ `0338` ◌̸
`2248` ≈ Almost Equal To
=asymptotic to
`2249` ≉ Not Almost Equal To
`2248` ≈ `0338` ◌̸
`224A` ≊ Almost Equal Or Equal To
`224B` ≋ Triple Tilde
`224C` ≌ All Equal To
reversed tilde and lazy S are glyph variants
`224D` ≍ Equivalent To
`224E` ≎ Geometrically Equivalent To
`224F` ≏ Difference Between
`2250` ≐ Approaches The Limit
`2251` ≑ Geometrically Equal To
`2252` ≒ Approximately Equal To Or The Image Of
=nearly equals
`2253` ≓ Image Of Or Approximately Equal To
`2254` ≔ Colon Equals
`2255` ≕ Equals Colon
`2256` ≖ Ring In Equal To
`2257` ≗ Ring Equal To
=approximately equal to
`2258` ≘ Corresponds To
`2259` ≙ Estimates
=corresponds to
`225A` ≚ Equiangular To
`225B` ≛ Star Equals
`225C` ≜ Delta Equal To
=equiangular
=equal to by definition
`225D` ≝ Equal To By Definition
`225E` ≞ Measured By
`225F` ≟ Questioned Equal To
`2260` ≠ Not Equal To
`003D` = equals sign
`01C2` ǂ latin letter alveolar click
`003D` = `0338` ◌̸
`2261` ≡ Identical To
`2262` ≢ Not Identical To
`2261` ≡ `0338` ◌̸
`2263` ≣ Strictly Equivalent To
`2264` ≤ Less-Than Or Equal To
`2265` ≥ Greater-Than Or Equal To
`2266` ≦ Less-Than Over Equal To
`2267` ≧ Greater-Than Over Equal To
`2268` ≨ Less-Than But Not Equal To
`2269` ≩ Greater-Than But Not Equal To
`226A` ≪ Much Less-Than
`00AB` « left-pointing double angle quotation mark
`226B` ≫ Much Greater-Than
`00BB` » right-pointing double angle quotation mark
`226C` ≬ Between
=plaintiff, quantic
`226D` ≭ Not Equivalent To
`224D` ≍ `0338` ◌̸
`226E` ≮ Not Less-Than
`003C` < `0338` ◌̸
`226F` ≯ Not Greater-Than
`003E` > `0338` ◌̸
`2270` ≰ Neither Less-Than Nor Equal To
`2264` ≤ `0338` ◌̸
`2271` ≱ Neither Greater-Than Nor Equal To
`2265` ≥ `0338` ◌̸
`2272` ≲ Less-Than Or Equivalent To
`2273` ≳ Greater-Than Or Equivalent To
`2274` ≴ Neither Less-Than Nor Equivalent To
`2272` ≲ `0338` ◌̸
`2275` ≵ Neither Greater-Than Nor Equivalent To
`2273` ≳ `0338` ◌̸
`2276` ≶ Less-Than Or Greater-Than
`2277` ≷ Greater-Than Or Less-Than
`2278` ≸ Neither Less-Than Nor Greater-Than
`2276` ≶ `0338` ◌̸
`2279` ≹ Neither Greater-Than Nor Less-Than
`2277` ≷ `0338` ◌̸
`227A` ≺ Precedes
=lower rank than
`22B0` ⊰ precedes under relation
`227B` ≻ Succeeds
=higher rank than
`22B1` ⊱ succeeds under relation
`227C` ≼ Precedes Or Equal To
`227D` ≽ Succeeds Or Equal To
`227E` ≾ Precedes Or Equivalent To
`227F` ≿ Succeeds Or Equivalent To
`2280` ⊀ Does Not Precede
`227A` ≺ `0338` ◌̸
`2281` ⊁ Does Not Succeed
`227B` ≻ `0338` ◌̸
`2282` ⊂ Subset Of
=included in set
`2E26` ⸦ left sideways u bracket
`2283` ⊃ Superset Of
=includes in set
`2E27` ⸧ right sideways u bracket
`2284` ⊄ Not A Subset Of
`2282` ⊂ `0338` ◌̸
`2285` ⊅ Not A Superset Of
`2283` ⊃ `0338` ◌̸
`2286` ⊆ Subset Of Or Equal To
`2287` ⊇ Superset Of Or Equal To
`2288` ⊈ Neither A Subset Of Nor Equal To
`2286` ⊆ `0338` ◌̸
`2289` ⊉ Neither A Superset Of Nor Equal To
`2287` ⊇ `0338` ◌̸
`228A` ⊊ Subset Of With Not Equal To
`228B` ⊋ Superset Of With Not Equal To

## Operators

`228C` ⊌ Multiset
`228D` ⊍ Multiset Multiplication
`2A03` ⨃ n-ary union operator with dot
`2A40` ⩀ intersection with dot
`228E` ⊎ Multiset Union
`2A04` ⨄ n-ary union operator with plus

## Relations

`228F` ⊏ Square Image Of
`2290` ⊐ Square Original Of
`2291` ⊑ Square Image Of Or Equal To
`2292` ⊒ Square Original Of Or Equal To

## Operators

`2293` ⊓ Square Cap
`2A05` ⨅ n-ary square intersection operator
`2294` ⊔ Square Cup
`2A06` ⨆ n-ary square union operator
`2295` ⊕ Circled Plus
=direct sum
=vector pointing into page
`2A01` ⨁ n-ary circled plus operator
`1F728` 🜨 alchemical symbol for verdigris
`2296` ⊖ Circled Minus
=symmetric difference
`29B5` ⦵ circle with horizontal bar
`1F714` 🜔 alchemical symbol for salt
`2297` ⊗ Circled Times
=tensor product
=vector pointing into page
`26D2` ⛒ circled crossing lanes
`2A02` ⨂ n-ary circled times operator
`2BBE` ⮾ circled x
`2298` ⊘ Circled Division Slash
`2299` ⊙ Circled Dot Operator
=direct product
=vector pointing out of page
`0298` ʘ latin letter bilabial click
`2609` ☉ sun
`2A00` ⨀ n-ary circled dot operator
`229A` ⊚ Circled Ring Operator
`233E` ⌾ apl functional symbol circle jot
`25CE` ◎ bullseye
`229B` ⊛ Circled Asterisk Operator
`235F` ⍟ apl functional symbol circle star
`229C` ⊜ Circled Equals
`229D` ⊝ Circled Dash
`229E` ⊞ Squared Plus
`229F` ⊟ Squared Minus
`22A0` ⊠ Squared Times
`2612` ☒ ballot box with x
`26DD` ⛝ squared saltire
`1F771` 🝱 alchemical symbol for month
`22A1` ⊡ Squared Dot Operator
`1F755` 🝕 alchemical symbol for urine
`22A2` ⊢ Right Tack
=turnstile
=proves, implies, yields
=reducible
`22A3` ⊣ Left Tack
=reverse turnstile
=non-theorem, does not yield

## Miscellaneous mathematical symbols

`22A4` ⊤ Down Tack
=top
`2E06` ⸆ raised interpolation marker
`1F768` 🝨 alchemical symbol for crucible-4
`22A5` ⊥ Up Tack
=base, bottom
`27C2` ⟂ perpendicular

## Relations

`22A6` ⊦ Assertion
=reduces to
`22A7` ⊧ Models
`22A8` ⊨ True
=statement is true, valid
=is a tautology
=satisfies
=results in
`22A9` ⊩ Forces
`22AA` ⊪ Triple Vertical Bar Right Turnstile
`22AB` ⊫ Double Vertical Bar Double Right Turnstile
`22AC` ⊬ Does Not Prove
`22A2` ⊢ `0338` ◌̸
`22AD` ⊭ Not True
`22A8` ⊨ `0338` ◌̸
`22AE` ⊮ Does Not Force
`22A9` ⊩ `0338` ◌̸
`22AF` ⊯ Negated Double Vertical Bar Double Right Turnstile
`22AB` ⊫ `0338` ◌̸
`22B0` ⊰ Precedes Under Relation
`227A` ≺ precedes
`22B1` ⊱ Succeeds Under Relation
`227B` ≻ succeeds
`22B2` ⊲ Normal Subgroup Of
`25C5` ◅ white left-pointing pointer
`22B3` ⊳ Contains As Normal Subgroup
`25BB` ▻ white right-pointing pointer
`22B4` ⊴ Normal Subgroup Of Or Equal To
`22B5` ⊵ Contains As Normal Subgroup Or Equal To
`22B6` ⊶ Original Of
`22B7` ⊷ Image Of
`22B8` ⊸ Multimap
`22B9` ⊹ Hermitian Conjugate Matrix

## Operators

`22BA` ⊺ Intercalate
`22BB` ⊻ Xor
`26BA` ⚺ semisextile
`22BC` ⊼ Nand
`2305` ⌅ projective
`26BB` ⚻ quincunx
`22BD` ⊽ Nor

## Miscellaneous mathematical symbols

`22BE` ⊾ Right Angle With Arc
`22BF` ⊿ Right Triangle

## N-ary operators

`22C0` ⋀ N-Ary Logical And
also used to denote the universal quantifier
`2227` ∧ logical and
`22C1` ⋁ N-Ary Logical Or
also used to denote the existential quantifier
`2228` ∨ logical or
`22C2` ⋂ N-Ary Intersection
=z notation generalised intersection
`2229` ∩ intersection
`22C3` ⋃ N-Ary Union
=z notation generalised union
`222A` ∪ union

## Operators

`22C4` ⋄ Diamond Operator
`25C7` ◇ white diamond
`2B29` ⬩ black small diamond
`22C5` ⋅ Dot Operator
preferred to `00B7` · for denotation of multiplication
`00B7` · middle dot
`22C6` ⋆ Star Operator
APL
`066D` ‎٭‎ arabic five pointed star
`2605` ★ black star
`2B50` ⭐ white medium star
`22C7` ⋇ Division Times

## Relation

`22C8` ⋈ Bowtie
`2445` ⑅ ocr bow tie

## Operators

`22C9` ⋉ Left Normal Factor Semidirect Product
`22CA` ⋊ Right Normal Factor Semidirect Product
`22CB` ⋋ Left Semidirect Product
`22CC` ⋌ Right Semidirect Product
`2041` ⁁ caret insertion point

## Relation

`22CD` ⋍ Reversed Tilde Equals

## Logical operators

`22CE` ⋎ Curly Logical Or
`22CF` ⋏ Curly Logical And

## Relations

`22D0` ⋐ Double Subset
`22D1` ⋑ Double Superset

## Operators

`22D2` ⋒ Double Intersection
`22D3` ⋓ Double Union

## Relations

`22D4` ⋔ Pitchfork
=proper intersection
`22D5` ⋕ Equal And Parallel To
`2317` ⌗ viewdata square
`22D6` ⋖ Less-Than With Dot
`22D7` ⋗ Greater-Than With Dot
`22D8` ⋘ Very Much Less-Than
`22D9` ⋙ Very Much Greater-Than
`22DA` ⋚ Less-Than Equal To Or Greater-Than
`22DB` ⋛ Greater-Than Equal To Or Less-Than
`22DC` ⋜ Equal To Or Less-Than
`22DD` ⋝ Equal To Or Greater-Than
`22DE` ⋞ Equal To Or Precedes
`22DF` ⋟ Equal To Or Succeeds
`22E0` ⋠ Does Not Precede Or Equal
`227C` ≼ `0338` ◌̸
`22E1` ⋡ Does Not Succeed Or Equal
`227D` ≽ `0338` ◌̸
`22E2` ⋢ Not Square Image Of Or Equal To
`2291` ⊑ `0338` ◌̸
`22E3` ⋣ Not Square Original Of Or Equal To
`2292` ⊒ `0338` ◌̸
`22E4` ⋤ Square Image Of Or Not Equal To
`22E5` ⋥ Square Original Of Or Not Equal To
`22E6` ⋦ Less-Than But Not Equivalent To
`22E7` ⋧ Greater-Than But Not Equivalent To
`22E8` ⋨ Precedes But Not Equivalent To
`22E9` ⋩ Succeeds But Not Equivalent To
`22EA` ⋪ Not Normal Subgroup Of
`22B2` ⊲ `0338` ◌̸
`22EB` ⋫ Does Not Contain As Normal Subgroup
`22B3` ⊳ `0338` ◌̸
`22EC` ⋬ Not Normal Subgroup Of Or Equal To
`22B4` ⊴ `0338` ◌̸
`22ED` ⋭ Does Not Contain As Normal Subgroup Or Equal
`22B5` ⊵ `0338` ◌̸
`22EE` ⋮ Vertical Ellipsis
these four ellipses are used for matrix row/column elision
`205D` ⁝ tricolon
`2026` … horizontal ellipsis
`FE19` ︙ presentation form for vertical horizontal ellipsis
`22EF` ⋯ Midline Horizontal Ellipsis
`22F0` ⋰ Up Right Diagonal Ellipsis
`22F1` ⋱ Down Right Diagonal Ellipsis
`22F2` ⋲ Element Of With Long Horizontal Stroke
`22F3` ⋳ Element Of With Vertical Bar At End Of Horizontal Stroke
`22F4` ⋴ Small Element Of With Vertical Bar At End Of Horizontal Stroke
`22F5` ⋵ Element Of With Dot Above
`22F6` ⋶ Element Of With Overbar
`22F7` ⋷ Small Element Of With Overbar
`22F8` ⋸ Element Of With Underbar
`22F9` ⋹ Element Of With Two Horizontal Strokes
`22FA` ⋺ Contains With Long Horizontal Stroke
`22FB` ⋻ Contains With Vertical Bar At End Of Horizontal Stroke
`22FC` ⋼ Small Contains With Vertical Bar At End Of Horizontal Stroke
`22FD` ⋽ Contains With Overbar
`22FE` ⋾ Small Contains With Overbar
`22FF` ⋿ Z Notation Bag Membership