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
  =gradient, del
  •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
  •may be used to represent the telephony asterisk seen on keypads
  →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
  =radical sign
  →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
  =with (in chess notation)
  →2BFE ⯾ reversed right angle
2220 ∠ Angle
  •used in astrological contexts for aspect semisquare
  →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
  →27D1 ⟑ and with dot
2228 ∨ Logical Or
  =vee, disjunction
  →22C1 ⋁ n-ary logical or
  →2304 ⌄ down arrowhead
  →27C7 ⟇ or with dot inside
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
  →2A70 ⩰ approximately equal or 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
  →003C < less-than sign
2265 ≥ Greater-Than Or Equal To
  →003E > greater-than sign
2266 ≦ Less-Than Over Equal To
2267 ≧ Greater-Than Over Equal To
2268 ≨ Less-Than But Not Equal To
  →2A87 ⪇ less-than and single-line not equal to
2269 ≩ Greater-Than But Not Equal To
  →2A88 ⪈ greater-than and single-line 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
  =z notation bag addition
  →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
  →1F774 🝴 lot of fortune
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
  =NoDerivatives (symbol used for Creative Commons licenses)
  →1F16D 🅭 circled cc
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
  →1FBBC 🮼 right open squared dot
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
  →00D7 × multiplication sign
  →00F7 ÷ division sign
  →2715 ✕ multiplication x

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 ◌̸

Matrix ellipses

These four ellipsis characters are used for matrix row/column elision.
22EE ⋮ Vertical Ellipsis
  →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

Relations

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

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