Copy a discrete math symbol, or read off its Alt code, HTML name and LaTeX macro, topic by topic.
Discrete Math Symbols Alt Codes
¬Not signU+00ACReady to copy
84 symbols and notations. Tap any value to copy it.
84 symbols and notations. A character and its name both copy the character. A code copies the code.
Character
Name
Code point
Alt code
Option keys
HTML
LaTeX
U+00AC. Negation, read not: ¬p is true when p is false. Alt+170 or Alt+0172 on a number pad, Option+L, ¬ in HTML and \neg in LaTeX.
U+2227. Conjunction, read and: p ∧ q is true when p and q are both true. Unicode also calls it wedge. ∧ in HTML, \wedge or \land in LaTeX.
U+2228. Disjunction, read or: p ∨ q is true when p, q or both are true. Unicode also calls it vee. ∨ in HTML, \vee or \lor in LaTeX.
U+2192. The conditional, read if p then q, or p implies q. The same arrow joins the domain of a function to its codomain. Alt+26, → in HTML, \to or \rightarrow in LaTeX.
U+2194. The biconditional, read if and only if: p ↔ q is true when p and q are both true or both false. Alt+29, ↔ in HTML and \leftrightarrow in LaTeX.
U+2200. The universal quantifier, read for all: ∀x P(x) says that every x has the property P. ∀ in HTML and \forall in LaTeX.
U+2203. The existential quantifier: ∃x P(x) says that at least one x has the property P. ∃ in HTML and \exists in LaTeX.
Made of U+2203 and an exclamation mark, U+0021. Uniqueness: ∃!x P(x) says that exactly one x has the property P. \exists followed by ! in LaTeX.
U+2295. Exclusive or, XOR: p ⊕ q is true when p or q is true and not both. ⊕ in HTML and \oplus in LaTeX.
U+2261. Between two statements, is logically equivalent to. Between two integers, congruence: a ≡ b (mod m). Alt+240, ≡ in HTML and \equiv in LaTeX.
U+21D2. Read implies. In written reasoning it has largely taken the place of →, which logic keeps for the conditional. ⇒ in HTML and \Rightarrow in LaTeX.
U+22A5. A statement that is always false, also written F. It is one of the signs set where a proof reaches a contradiction, too. ⊥ in HTML and \bot in LaTeX.
U+2234. Short for therefore: it stands before a conclusion. ∴ in HTML; \therefore in LaTeX needs amssymb.
U+220E. Marks the end of a proof; the letters Q.E.D. are often used for the same purpose. Unicode's other name for it is q.e.d.
U+21AF. One of the signs that mark a contradiction reached in a proof. No one sign is agreed on.
U+203B. Another sign in use for a contradiction in a proof.
Made of U+21D2 and U+21D0: two double arrows that collide, a third way to mark a contradiction. \Rightarrow and \Leftarrow in LaTeX.
U+2208. Set membership: x ∈ S says the set S has x as an element, read is in or belongs to. ∈ in HTML and \in in LaTeX.
U+2209. Read is not in: x ∉ S. ∉ in HTML and \notin in LaTeX.
U+2286. A ⊆ B says that A is a subset of B, and that the two may be equal. ⊆ in HTML and \subseteq in LaTeX.
U+2282. Set inclusion. Some books mean any subset by it and others only a proper subset. ⊂ in HTML and \subset in LaTeX.
U+222A. A ∪ B is the set that holds every element of A and of B. ∪ in HTML and \cup in LaTeX.
U+2229. A ∩ B is the set of the elements found in both A and B. Alt+239, ∩ in HTML and \cap in LaTeX.
U+2205. The set with no elements, also written {}. ∅ in HTML and \emptyset in LaTeX.
U+2216. A ∖ B is the set of the elements of A that are not in B. ∖ in HTML and \setminus in LaTeX.
U+00D7. Between two sets, the Cartesian product: A × B is the set of every pair made of one element of A and one of B. Alt+0215, × in HTML and \times in LaTeX.
U+1D4AB. Unicode's note: power set. 𝒫(S) collects every subset of S, also written P(S). 𝒫 in HTML.
Made of the set's name between two vertical lines, U+007C. The number of elements in A.
Made of A and U+1D9C, a raised small c. The set of the elements that are not in A. An overline over the A is another way to write it.
U+2115. The natural numbers: 1, 2, 3 and so on, or in some books 0, 1, 2 and so on. ℕ in HTML.
U+2124. The integers: the whole numbers, negative, zero and positive. ℤ in HTML.
Made of A, U+2192 and B. A function with domain A and codomain B; f : A → B names it f.
U+21A6. Read maps to: x ↦ x² defines the squaring function without naming it. ↦ in HTML and \mapsto in LaTeX.
U+2218. Function composition: (g ∘ f)(x) = g(f(x)). Unicode's other name for it is composite function. ∘ in HTML and \circ in LaTeX.
Made of U+230A, x and U+230B. The largest integer no greater than x. ⌊ and ⌋ in HTML, \lfloor and \rfloor in LaTeX.
Made of U+2308, x and U+2309. The lowest integer that is not less than x. ⌈ and ⌉ in HTML, \lceil and \rceil in LaTeX.
Made of f, U+207B and U+00B9, a raised minus one. f⁻¹(Y) is the preimage of the set Y under f.
U+21A3. Unicode's note: z notation total injection. An injection is a one-to-one function. ↣ in HTML; \rightarrowtail in LaTeX needs amssymb.
U+21A0. Unicode's note: z notation total surjection. A surjection is an onto function. ↠ in HTML; \twoheadrightarrow in LaTeX needs amssymb.
U+2916. Unicode's note: bijective mapping. A bijection is one-to-one and onto. ⤖ in HTML.
U+223C. A tilde between two things is the standard notation for an equivalence relation: a ∼ b. ∼ in HTML and \sim in LaTeX.
Made of a in square brackets. Under an equivalence relation, the class of everything equivalent to a.
U+227C. Named precedes or equal to: ≤ drawn with curved strokes. ≼ in HTML; \preccurlyeq in LaTeX needs amssymb.
Made of n and an exclamation mark, U+0021. The first n positive integers, multiplied. As a rate of growth, O(n!) is the factorial order.
Made of U+2099, P and U+1D63, a subscript n and r. The number of ordered arrangements of r things out of n.
Made of U+2099, C and U+1D63. The number of selections of r things out of n when the order of selection does not matter.
U+2211. Unicode's summation sign: the sum of the terms that its subscript and superscript mark out. Option+W, ∑ in HTML and \sum in LaTeX.
U+220F. Unicode's product sign: the product of the terms that its subscript and superscript mark out. Option+Shift+P, ∏ in HTML and \prod in LaTeX.
U+2223. a ∣ b says that a divides b evenly. Unicode's other name for it is such that. ∣ in HTML and \mid in LaTeX.
U+2224. a ∤ b says that a is not a divisor of b. ∤ in HTML; \nmid in LaTeX needs amssymb.
Made of the letters g, c and d. gcd(a,b) is the largest positive integer that divides both a and b: gcd(8,12) is 4.
Made of the letters l, c and m. lcm(a,b) is the least positive integer both a and b divide.
Made of the letters m, o and d. b mod m is the integer r from 0 up to m − 1 with r ≡ b (mod m): the remainder.
Made of U+2124 and U+2099, a subscript n. The integers modulo n, also written ℤ/nℤ.
Made of U+03C6, the Greek letter phi, and n in brackets. Counts the positive integers up to n that are coprime to n.
Made of x and U+2032, the prime. One way to write NOT x. An overline over the x is another, used primarily in electronics.
U+00B7. AND, written x·y. OR has a matching form with the plus sign, x + y. Alt+250 or Alt+0183, Option+Shift+9 and · in HTML.
U+2191. The Sheffer stroke, NAND: p ↑ q is the negation of p ∧ q. Alt+24, ↑ in HTML and \uparrow in LaTeX.
U+2193. NOR, the Peirce arrow: p ↓ q is true only when p and q are both false. Alt+25, ↓ in HTML and \downarrow in LaTeX.
Made of K and U+2099, a subscript n. The graph in which every two of the n vertices are adjacent.
Made of K, U+2098, a comma and U+2099. m vertices on one side and n on the other, with every two vertices on opposite sides adjacent.
Made of C and U+2099. The cycle graph with n vertices.
Made of P and U+2099. The path graph: n vertices in a row, each joined to the next.
Made of Q and U+2099. The graph made of the n-dimensional hypercube's vertices and edges.
Made of W and U+2099. A cycle plus one more vertex joined to every vertex of it. The numbering of wheels is not standard.
Made of the letters d, e and g. deg(v) counts the edges that meet the vertex v.
Made of U+0394, the capital delta, and G in brackets. The largest degree of any vertex of G.
Made of U+03B4, the small delta, and G in brackets. The smallest degree of any vertex of G.
Made of U+03C7, the Greek letter chi, and G in brackets. The minimum number of colours needed in a proper colouring of G.
Made of U+03C7 and a prime, U+2032. χ′(G) is the minimum number of colours needed in a proper edge colouring of G.
Made of U+03C9, the small omega, and G in brackets. The largest clique's order in G.
Made of U+03B1, the small alpha, and G in brackets. The size of the largest set of vertices with no edge between any two.
Made of V between two vertical lines, U+007C. The number of vertices.
Made of E between two vertical lines, U+007C. The number of edges.
Made of a capital O and n in brackets. f = O(g) says that f is bounded above by g, up to a constant factor. O(n) is linear growth.
Made of U+03A9 and n in brackets. f = Ω(g) says that g, times some constant, is a lower bound on f.
Made of U+0398 and n in brackets. f = Θ(g) says that g bounds f both above and below, up to constant factors.
Made of a small o and n in brackets. f = o(g) says that g asymptotically dominates f.
Made of U+03C9 and n in brackets. f = ω(g) says that f dominates g asymptotically.
U+0398. The letter of big theta. Alt+233, Θ in HTML and \Theta in LaTeX.
U+03A9. The letter of big omega. Alt+234, Option+Z, Ω in HTML and \Omega in LaTeX.
Made of a capital O and 1 in brackets. The constant order: any bounded function is O(1).
Made of n and U+00B2, superscript two. O(n²) is the quadratic order.
Made of 2 and U+207F, superscript n. O(2ⁿ) is an exponential order.
Eleven of these discrete math symbols have Alt codes: ¬ is Alt+170, → Alt+26, ↔ Alt+29, ≡ Alt+240, ∩ Alt+239, × Alt+0215, · Alt+250, ↑ Alt+24, ↓ Alt+25, Θ Alt+233 and Ω Alt+234. Copy the rest.
Read the full guide
A chart in nine topics
The chart keeps the symbols used in discrete math in topics: logic, proofs, set theory, functions and relations, counting, number theory, Boolean algebra, graphs and big O. Notation of more than one character, such as n!, ⌊x⌋, ₙCᵣ, Kₙ and O(n), is a row that pastes whole.
Function and relation symbols
A→B is a function with domain A and codomain B, ↦ reads maps to, ∘ composes two functions, and ⌊x⌋ and ⌈x⌉ are floor and ceiling. Among relations, ∼ is the standard sign of an equivalence relation and ≡ is congruence modulo m.
∣ for divides is U+2223, a character apart from the vertical line of a keyboard, U+007C.
On paper the table is a black and white symbols sheet with every code column on it.
Questions and answers
What is symbolic notation in discrete math?
Statements written in signs in place of words: ¬ for not, ∧ for and, ∨ for or, → for if then, ↔ for if and only if, ∀ for for all and ∃ for there exists. ∀x P(x) says every x has the property P.
Which proof symbols does discrete math use?
∴ is short for therefore and ⇒ reads implies. ∎ marks the end of a proof, in place of Q.E.D. A contradiction has no agreed sign: ↯, ※, ⊥ and ⇒⇐ are all in use.
What do the set theory symbols mean?
∈ is membership, ⊆ subset, ∪ union, ∩ intersection, ∖ difference, × the Cartesian product and ∅ the empty set. |A| is the number of elements in A, and 𝒫(A) is its power set, the set of all its subsets.
Is it discrete, discreet or descrete math?
Discrete: the math of discrete structures, such as integers, graphs and statements in logic, rather than continuous ones. Discreet sounds the same and means respectful of privacy. Descrete is a misspelling.
All 84 symbols and notations
Start here 7
∀For allU+2200. The universal quantifier, read for all: ∀x P(x) says that every x has the property P. ∀ in HTML and \forall in LaTeX.
↔Left right arrowU+2194. The biconditional, read if and only if: p ↔ q is true when p and q are both true or both false. Alt+29, ↔ in HTML and \leftrightarrow in LaTeX.
∧Logical andU+2227. Conjunction, read and: p ∧ q is true when p and q are both true. Unicode also calls it wedge. ∧ in HTML, \wedge or \land in LaTeX.
∨Logical orU+2228. Disjunction, read or: p ∨ q is true when p, q or both are true. Unicode also calls it vee. ∨ in HTML, \vee or \lor in LaTeX.
¬Not signU+00AC. Negation, read not: ¬p is true when p is false. Alt+170 or Alt+0172 on a number pad, Option+L, ¬ in HTML and \neg in LaTeX.
→Rightwards arrowU+2192. The conditional, read if p then q, or p implies q. The same arrow joins the domain of a function to its codomain. Alt+26, → in HTML, \to or \rightarrow in LaTeX.
∃There existsU+2203. The existential quantifier: ∃x P(x) says that at least one x has the property P. ∃ in HTML and \exists in LaTeX.
Show all 84 symbols and notations (77 more)
Logic 5
⊕Circled plusU+2295. Exclusive or, XOR: p ⊕ q is true when p or q is true and not both. ⊕ in HTML and \oplus in LaTeX.
≡Identical toU+2261. Between two statements, is logically equivalent to. Between two integers, congruence: a ≡ b (mod m). Alt+240, ≡ in HTML and \equiv in LaTeX.
⇒Rightwards double arrowU+21D2. Read implies. In written reasoning it has largely taken the place of →, which logic keeps for the conditional. ⇒ in HTML and \Rightarrow in LaTeX.
∃!There exists exactly oneMade of U+2203 and an exclamation mark, U+0021. Uniqueness: ∃!x P(x) says that exactly one x has the property P. \exists followed by ! in LaTeX.
⊥Up tackU+22A5. A statement that is always false, also written F. It is one of the signs set where a proof reaches a contradiction, too. ⊥ in HTML and \bot in LaTeX.
Proofs 5
⇒⇐Contradiction arrowsMade of U+21D2 and U+21D0: two double arrows that collide, a third way to mark a contradiction. \Rightarrow and \Leftarrow in LaTeX.
↯Downwards zigzag arrowU+21AF. One of the signs that mark a contradiction reached in a proof. No one sign is agreed on.
∎End of proofU+220E. Marks the end of a proof; the letters Q.E.D. are often used for the same purpose. Unicode's other name for it is q.e.d.
※Reference markU+203B. Another sign in use for a contradiction in a proof.
∴ThereforeU+2234. Short for therefore: it stands before a conclusion. ∴ in HTML; \therefore in LaTeX needs amssymb.
Sets 14
|A|Cardinality of AMade of the set's name between two vertical lines, U+007C. The number of elements in A.
AᶜComplement of AMade of A and U+1D9C, a raised small c. The set of the elements that are not in A. An overline over the A is another way to write it.
ℕDouble-struck capital nU+2115. The natural numbers: 1, 2, 3 and so on, or in some books 0, 1, 2 and so on. ℕ in HTML.
ℤDouble-struck capital zU+2124. The integers: the whole numbers, negative, zero and positive. ℤ in HTML.
∈Element ofU+2208. Set membership: x ∈ S says the set S has x as an element, read is in or belongs to. ∈ in HTML and \in in LaTeX.
∅Empty setU+2205. The set with no elements, also written {}. ∅ in HTML and \emptyset in LaTeX.
∩IntersectionU+2229. A ∩ B is the set of the elements found in both A and B. Alt+239, ∩ in HTML and \cap in LaTeX.
𝒫Mathematical script capital pU+1D4AB. Unicode's note: power set. 𝒫(S) collects every subset of S, also written P(S). 𝒫 in HTML.
×Multiplication signU+00D7. Between two sets, the Cartesian product: A × B is the set of every pair made of one element of A and one of B. Alt+0215, × in HTML and \times in LaTeX.
∉Not an element ofU+2209. Read is not in: x ∉ S. ∉ in HTML and \notin in LaTeX.
∖Set minusU+2216. A ∖ B is the set of the elements of A that are not in B. ∖ in HTML and \setminus in LaTeX.
⊂Subset ofU+2282. Set inclusion. Some books mean any subset by it and others only a proper subset. ⊂ in HTML and \subset in LaTeX.
⊆Subset of or equal toU+2286. A ⊆ B says that A is a subset of B, and that the two may be equal. ⊆ in HTML and \subseteq in LaTeX.
∪UnionU+222A. A ∪ B is the set that holds every element of A and of B. ∪ in HTML and \cup in LaTeX.
Functions and relations 12
⌈x⌉Ceiling of xMade of U+2308, x and U+2309. The lowest integer that is not less than x. ⌈ and ⌉ in HTML, \lceil and \rceil in LaTeX.
[a]Equivalence class of aMade of a in square brackets. Under an equivalence relation, the class of everything equivalent to a.
⌊x⌋Floor of xMade of U+230A, x and U+230B. The largest integer no greater than x. ⌊ and ⌋ in HTML, \lfloor and \rfloor in LaTeX.
A→BFunction from A to BMade of A, U+2192 and B. A function with domain A and codomain B; f : A → B names it f.
≼Precedes or equal toU+227C. Named precedes or equal to: ≤ drawn with curved strokes. ≼ in HTML; \preccurlyeq in LaTeX needs amssymb.
f⁻¹Preimage under fMade of f, U+207B and U+00B9, a raised minus one. f⁻¹(Y) is the preimage of the set Y under f.
↦Rightwards arrow from barU+21A6. Read maps to: x ↦ x² defines the squaring function without naming it. ↦ in HTML and \mapsto in LaTeX.
↣Rightwards arrow with tailU+21A3. Unicode's note: z notation total injection. An injection is a one-to-one function. ↣ in HTML; \rightarrowtail in LaTeX needs amssymb.
↠Rightwards two headed arrowU+21A0. Unicode's note: z notation total surjection. A surjection is an onto function. ↠ in HTML; \twoheadrightarrow in LaTeX needs amssymb.
⤖Rightwards two-headed arrow with tailU+2916. Unicode's note: bijective mapping. A bijection is one-to-one and onto. ⤖ in HTML.
∘Ring operatorU+2218. Function composition: (g ∘ f)(x) = g(f(x)). Unicode's other name for it is composite function. ∘ in HTML and \circ in LaTeX.
∼Tilde operatorU+223C. A tilde between two things is the standard notation for an equivalence relation: a ∼ b. ∼ in HTML and \sim in LaTeX.
Counting 5
ₙCᵣCombinations of n taken rMade of U+2099, C and U+1D63. The number of selections of r things out of n when the order of selection does not matter.
n!n factorialMade of n and an exclamation mark, U+0021. The first n positive integers, multiplied. As a rate of growth, O(n!) is the factorial order.
∏N-ary productU+220F. Unicode's product sign: the product of the terms that its subscript and superscript mark out. Option+Shift+P, ∏ in HTML and \prod in LaTeX.
∑N-ary summationU+2211. Unicode's summation sign: the sum of the terms that its subscript and superscript mark out. Option+W, ∑ in HTML and \sum in LaTeX.
ₙPᵣPermutations of n taken rMade of U+2099, P and U+1D63, a subscript n and r. The number of ordered arrangements of r things out of n.
Number theory 7
∣DividesU+2223. a ∣ b says that a divides b evenly. Unicode's other name for it is such that. ∣ in HTML and \mid in LaTeX.
∤Does not divideU+2224. a ∤ b says that a is not a divisor of b. ∤ in HTML; \nmid in LaTeX needs amssymb.
φ(n)Euler's totient functionMade of U+03C6, the Greek letter phi, and n in brackets. Counts the positive integers up to n that are coprime to n.
gcdGreatest common divisorMade of the letters g, c and d. gcd(a,b) is the largest positive integer that divides both a and b: gcd(8,12) is 4.
ℤₙIntegers modulo nMade of U+2124 and U+2099, a subscript n. The integers modulo n, also written ℤ/nℤ.
lcmLeast common multipleMade of the letters l, c and m. lcm(a,b) is the least positive integer both a and b divide.
modModuloMade of the letters m, o and d. b mod m is the integer r from 0 up to m − 1 with r ≡ b (mod m): the remainder.
Boolean algebra 4
x′Complement of xMade of x and U+2032, the prime. One way to write NOT x. An overline over the x is another, used primarily in electronics.
↓Downwards arrowU+2193. NOR, the Peirce arrow: p ↓ q is true only when p and q are both false. Alt+25, ↓ in HTML and \downarrow in LaTeX.
·Middle dotU+00B7. AND, written x·y. OR has a matching form with the plus sign, x + y. Alt+250 or Alt+0183, Option+Shift+9 and · in HTML.
↑Upwards arrowU+2191. The Sheffer stroke, NAND: p ↑ q is the negation of p ∧ q. Alt+24, ↑ in HTML and \uparrow in LaTeX.
Graphs 15
χ′Chromatic indexMade of U+03C7 and a prime, U+2032. χ′(G) is the minimum number of colours needed in a proper edge colouring of G.
χ(G)Chromatic number of GMade of U+03C7, the Greek letter chi, and G in brackets. The minimum number of colours needed in a proper colouring of G.
ω(G)Clique number of GMade of U+03C9, the small omega, and G in brackets. The largest clique's order in G.
Kₘ,ₙComplete bipartite graphMade of K, U+2098, a comma and U+2099. m vertices on one side and n on the other, with every two vertices on opposite sides adjacent.
KₙComplete graph on n verticesMade of K and U+2099, a subscript n. The graph in which every two of the n vertices are adjacent.
CₙCycle graph on n verticesMade of C and U+2099. The cycle graph with n vertices.
degDegree of a vertexMade of the letters d, e and g. deg(v) counts the edges that meet the vertex v.
QₙHypercube graphMade of Q and U+2099. The graph made of the n-dimensional hypercube's vertices and edges.
α(G)Independence number of GMade of U+03B1, the small alpha, and G in brackets. The size of the largest set of vertices with no edge between any two.
Δ(G)Maximum degree of GMade of U+0394, the capital delta, and G in brackets. The largest degree of any vertex of G.
δ(G)Minimum degree of GMade of U+03B4, the small delta, and G in brackets. The smallest degree of any vertex of G.
|V|Order of a graphMade of V between two vertical lines, U+007C. The number of vertices.
PₙPath graph on n verticesMade of P and U+2099. The path graph: n vertices in a row, each joined to the next.
|E|Size of a graphMade of E between two vertical lines, U+007C. The number of edges.
WₙWheel graphMade of W and U+2099. A cycle plus one more vertex joined to every vertex of it. The numbering of wheels is not standard.
Big O 10
O(n)Big O of nMade of a capital O and n in brackets. f = O(g) says that f is bounded above by g, up to a constant factor. O(n) is linear growth.
Ω(n)Big omega of nMade of U+03A9 and n in brackets. f = Ω(g) says that g, times some constant, is a lower bound on f.
Θ(n)Big theta of nMade of U+0398 and n in brackets. f = Θ(g) says that g bounds f both above and below, up to constant factors.
O(1)Constant orderMade of a capital O and 1 in brackets. The constant order: any bounded function is O(1).
2ⁿExponential growthMade of 2 and U+207F, superscript n. O(2ⁿ) is an exponential order.
ΩGreek capital letter omegaU+03A9. The letter of big omega. Alt+234, Option+Z, Ω in HTML and \Omega in LaTeX.
ΘGreek capital letter thetaU+0398. The letter of big theta. Alt+233, Θ in HTML and \Theta in LaTeX.
o(n)Little o of nMade of a small o and n in brackets. f = o(g) says that g asymptotically dominates f.
ω(n)Little omega of nMade of U+03C9 and n in brackets. f = ω(g) says that f dominates g asymptotically.
n²Quadratic growthMade of n and U+00B2, superscript two. O(n²) is the quadratic order.