Mathematical Symbols
This document specifies the recommended notation for mathematical symbols in OI Wiki, along with some application examples.
This document is revised with reference to the symbol tables of GB/T 3102.11-1993, ISO 80000-2:2019, and Concrete Mathematics, thus it is largely compatible with the symbol systems commonly used in domestic textbooks and the customary notation in OI scenarios.
For LaTeX notation of symbols, please refer to the source code of this article
Mathematical Logic
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
| n1.1 | \(p \land q\) | conjunction of \(p\) and \(q\) | \(p\) and \(q\). |
| n1.2 | \(p \lor q\) | disjunction of \(p\) and \(q\) | \(p\) or \(q\); Here "or" is inclusive, meaning \(p \lor q\) is true if at least one of \(p\), \(q\) is a true statement. |
| n1.3 | \(\lnot p\) | negation of \(p\) | not \(p\). |
| n1.4 | \(p \implies q\) | \(p\) implies \(q\); if \(p\) is true, then \(q\) is true |
\(q \impliedby p\) is synonymous with \(p \implies q\). |
| n1.5 | \(p \iff q\) | \(p\) is equivalent to \(q\) | \((p \implies q) \land (q \implies p)\) is synonymous with \(p \iff q\). |
| n1.6 | \((\forall~x \in A)~~p(x)\) | for all \(x\) in \(A\), the proposition \(p(x)\) is true | If the set \(A\) under consideration is clear from context, the notation \((\forall~x)~~p(x)\) can be used. \(\forall\) is called the universal quantifier. For the meaning of \(x \in A\), see n2.1. |
| n1.7 | \((\exists~x \in A)~~p(x)\) | there exists an \(x\) in \(A\) such that \(p(x)\) is true | If the set \(A\) under consideration is clear from context, the notation \((\exists~x)~~p(x)\) can be used. \(\exists\) is called the existential quantifier. For the meaning of \(x \in A\), see n2.1. \((\exists!~x)~~p(x)\) (the uniqueness quantifier) means there is exactly one \(x\) such that \(p(x)\) is true. \(\exists!\) can also be written as \(\exists^1\). |
Set Theory
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
| n2.1 | \(x \in A\) | \(x\) belongs to \(A\); \(x\) is an element of the set \(A\) | \(A \ni x\) is synonymous with \(x \in A\). |
| n2.2 | \(y \notin A\) | \(y\) does not belong to \(A\); \(y\) is not an element of \(A\) | |
| n2.3 | \(\{x_1, x_2, \dots, x_n\}\) | set with elements \(x_1, x_2, \dots, x_n\) | Can also be written as \(\{x_i ~\vert~ i \in I\}\), where \(I\) denotes an index set. |
| n2.4 | \(\{x \in A ~\vert~ p(x)\}\) | set of all elements in \(A\) for which \(p(x)\) is true | Example: \(\{x \in \textbf{R} ~\vert~ x \geq 5\}\); If the set \(A\) under consideration is clear from context, the notation \(\{x ~\vert~ p(x)\}\) can be used (e.g., \(\{x ~\vert~ x \geq 5\}\) when only real numbers are considered). \(\vert\) can be replaced by a colon, e.g., \(\{x \in A : p(x)\}\). |
| n2.5 | \(\operatorname{card} A\); \(\vert A\vert\); \(\# A\) |
number of elements in \(A\), cardinality of \(A\) | |
| n2.6 | \(\varnothing\) | empty set | \(\emptyset\) should not be used. |
| n2.7 | \(B \subseteq A\) | \(B\) is included in \(A\); \(B\) is a subset of \(A\) | Every element of \(B\) belongs to \(A\). \(\subset\) can also be used with this meaning, but see the note in n2.8. \(A \supseteq B\) is synonymous with \(B \subseteq A\). |
| n2.8 | \(B \subset A\) | \(B\) is properly included in \(A\); \(B\) is a proper subset of \(A\) | Every element of \(B\) belongs to \(A\), and \(A\) contains at least one element not in \(B\). If \(\subset\) is used with the meaning of n2.7, then the symbol for n2.8 should be \(\subsetneq\). \(A \supset B\) is synonymous with \(B \subset A\). |
| n2.9 | \(A \cup B\) | union of \(A\) and \(B\) | \(A \cup B := \{x ~\vert~ x \in A \lor x \in B\}\); For the definition of \(:=\) see n4.3 |
| n2.10 | \(A \cap B\) | intersection of \(A\) and \(B\) | \(A \cap B := \{x ~\vert~ x \in A \land x \in B\}\); For the definition of \(:=\) see n4.3 |
| n2.11 | \(\displaystyle \bigcup\limits_{i=1}^n A_i\) | union of sets \(A_1, A_2, \dots, A_n\) | \(\displaystyle \bigcup\limits_{i=1}^n A_i=A_1\cup A_2\cup \dots \cup A_n\); Also \(\displaystyle \bigcup\nolimits_{i=1}^n\), \(\displaystyle \bigcup\limits_{i\in I}\), \(\displaystyle \bigcup\nolimits_{i\in I}\) can be used, where \(I\) denotes an index set; Furthermore, let \(P(i)\) be a proposition concerning \(i\), then \(\displaystyle \bigcup_{P(i)} A_i\) denotes the union of all \(A_i\) for which \(P(i)\) is true. |
| n2.12 | \(\displaystyle \bigcap\limits_{i=1}^n A_i\) | intersection of sets \(A_1, A_2, \dots, A_n\) | \(\displaystyle \bigcap\limits_{i=1}^n A_i=A_1\cap A_2\cap \dots \cap A_n\); Also \(\displaystyle \bigcap\nolimits_{i=1}^n\), \(\displaystyle \bigcap\limits_{i\in I}\), \(\displaystyle \bigcap\nolimits_{i\in I}\) can be used, where \(I\) denotes an index set; Furthermore, let \(P(i)\) be a proposition concerning \(i\), then \(\displaystyle \bigcap_{P(i)} A_i\) denotes the intersection of all \(A_i\) for which \(P(i)\) is true. |
| n2.13 | \(A \setminus B\) | difference of \(A\) and \(B\) | \(A \setminus B = \{x ~\vert~ x \in A \land x \notin B\}\); \(A - B\) should not be used; When \(B\) is a subset of \(A\), \(\complement_A B\) can also be used. If the set \(A\) is clear from context, \(A\) can be omitted. When no confusion arises, \(\overline{B}\) can also be used to denote the complement of set \(B\). |
| n2.14 | \((a, b)\) | ordered pair \(a\), \(b\); ordered couple \(a\), \(b\) |
\((a, b) = (c, d)\) if and only if \(a = c\) and \(b = d\). |
| n2.15 | \((a_1, a_2, \dots, a_n)\) | ordered \(n\)-tuple | See n2.14. |
| n2.16 | \(A \times B\) | Cartesian product of sets \(A\) and \(B\) | \(A \times B = \{(x, y) ~\vert~ x \in A \land y \in B\}\). |
| n2.17 | \(\displaystyle \prod\limits_{i=1}^{n} A_i\) | Cartesian product of sets \(A_1, A_2, \dots, A_n\) | \(\displaystyle \prod\limits_{i=1}^{n} A_i=\{(x_1, x_2, \dots, x_n) ~\vert~ x_1 \in A_1, x_2 \in A_2, \dots, x_n \in A_n\}\); \(A \times A \times \dots \times A\) is denoted \(A^n\), where \(n\) is the number of factors in the product; For another use of this symbol, see n6.8 |
| n2.18 | \(\mathrm{id}_A\) | diagonal set of \(A\times A\) | \(\mathrm{id}_A=\{(x, x)~\vert~x\in A\}\); If the set \(A\) is clear from context, \(A\) can be omitted. |
| n2.19 | \(\mathbf{1}_A\) | indicator function | \(\mathbf{1}_A(a)=[a\in A]\), where \([\cdot]\) is defined in n6.24. |
| n2.20 | \(\mathcal{P}(A)\); \(2^A\) |
power set | \(\mathcal{P}(A)=\{S:S\subseteq A\}\) |
Standard Number Sets and Intervals
Relations
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
| n4.1 | \(a = b\) | \(a\) equals \(b\) | \(\equiv\) is used to emphasize that an equality is an identity. Another meaning of this symbol is given in n4.18. |
| n4.2 | \(a \ne b\) | \(a\) is not equal to \(b\) | |
| n4.3 | \(a := b\) | \(a\) is defined as \(b\) | See n2.9,n2.10 |
| n4.4 | \(a \approx b\) | \(a\) is approximately equal to \(b\) | Equality is not excluded. |
| n4.5 | \(a \simeq b\) | \(a\) is asymptotically equal to \(b\) | Example: When \(x\to a\), \(\dfrac{1}{\sin(x-a)} \simeq \dfrac{1}{x-a}\); For the meaning of \(x \to a\), see n4.15. |
| n4.6 | \(a \propto b\) | \(a\) is proportional to \(b\) | \(a \sim b\) can also be used. \(\sim\) is also used for equivalence relations. |
| n4.7 | \(M \cong N\) | \(M\) is congruent to \(N\) | When \(M\) and \(N\) are point sets (geometric figures). This symbol is also used for isomorphism of algebraic structures. |
| n4.8 | \(a < b\) | \(a\) is less than \(b\) | |
| n4.9 | \(b > a\) | \(b\) is greater than \(a\) | |
| n4.10 | \(a \leq b\) | \(a\) is less than or equal to \(b\) | |
| n4.11 | \(b \geq a\) | \(b\) is greater than or equal to \(a\) | |
| n4.12 | \(a \ll b\) | \(a\) is much less than \(b\) | |
| n4.13 | \(b \gg a\) | \(b\) is much greater than \(a\) | |
| n4.14 | \(\infty\) | infinity | This symbol is not a number. \(+\infty\), \(-\infty\) can also be used. |
| n4.15 | \(x \to a\) | \(x\) tends to \(a\) | Usually appears in limit expressions. \(a\) can also be \(\infty\), \(+\infty\), \(-\infty\). |
| n4.16 | \(m \mid n\) | \(m\) divides \(n\) | For integers \(m\), \(n\): \((\exists~k \in \mathbf{Z})~~m\cdot k = n\). |
| n4.17 | \(m \perp n\) | \(m\) is coprime to \(n\) | For integers \(m\), \(n\): \((\nexists~k \in \mathbf{Z}_{>1})~~(k \mid m) \land (k \mid n)\); Another use of this symbol is given in n5.2 |
| n4.18 | \(n \equiv k \pmod m\) | \(n\) is congruent to \(k\) modulo \(m\) | For integers \(n\), \(k\), \(m\): \(m \mid (n - k)\); Do not confuse with the meaning mentioned in n4.1. |
Elementary Geometry
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
| n5.1 | \(\parallel\) | parallel | |
| n5.2 | \(\perp\) | perpendicular | Another use of this symbol is given in n4.17 |
| n5.3 | \(\angle\) | (plane) angle | |
| n5.4 | \(\overline{\mathrm{AB}}\) | line segment \(\mathrm{AB}\) | |
| n5.5 | \(\overrightarrow{\mathrm{AB}}\) | directed line segment \(\mathrm{AB}\) | |
| n5.6 | \(d(\mathrm{A}, \mathrm{B})\) | distance between points \(\mathrm{A}\) and \(\mathrm{B}\) | i.e., the length of \(\overline{\mathrm{AB}}\). |
Operators
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
| n6.1 | \(a + b\) | \(a\) plus \(b\) | |
| n6.2 | \(a - b\) | \(a\) minus \(b\) | |
| n6.3 | \(a \pm b\) | \(a\) plus or minus \(b\) | |
| n6.4 | \(a \mp b\) | \(a\) minus or plus \(b\) | \(-(a \pm b) = -a \mp b\). |
| n6.5 | \(a \cdot b\); \(a \times b\); \(ab\) |
\(a\) times \(b\) | If a decimal point appears, only \(\times\) should be used; Some use cases are given in n2.16,n2.17,n14.11,n14.12 |
| n6.6 | \(\dfrac{a}{b}\); \(a/b\); \(a:b\) |
\(a\) divided by \(b\) | \(\dfrac{a}{b}=a\cdot b^{-1}\); \(:\) can be used to denote the ratio of numerical values of the same dimension. \(÷\) should not be used. |
| n6.7 | \(\displaystyle \sum\limits_{i=1}^n a_i\) | \(a_1 + a_2 + \dots + a_n\) | Also \(\displaystyle \sum\nolimits_{i=1}^n a_i\), \(\displaystyle \sum\limits_i a_i\), \(\displaystyle \sum\nolimits_i a_i\), \(\displaystyle \sum a_i\) can be used; Let \(P(i)\) be a proposition concerning \(i\), then \(\displaystyle \sum_{P(i)} a_i\) denotes the sum of all \(a_i\) for which \(P(i)\) is true. |
| n6.8 | \(\displaystyle \prod\limits_{i=1}^n a_i\) | \(a_1 \cdot a_2 \cdot \dots \cdot a_n\) | Also \(\displaystyle \prod\nolimits_{i=1}^n a_i\), \(\displaystyle \prod\limits_i a_i\), \(\displaystyle \prod\nolimits_i a_i\), \(\displaystyle \prod a_i\) can be used; Let \(P(i)\) be a proposition concerning \(i\), then \(\displaystyle \prod_{P(i)} a_i\) denotes the product of all \(a_i\) for which \(P(i)\) is true; Another use of this symbol is given in n2.17 |
| n6.9 | \(a^p\) | \(a\) to the power \(p\) | |
| n6.10 | \(a^{1/2}\); \(\sqrt{a}\) |
\(a\) to the power \(1/2\), square root of \(a\) | \(\sqrt{}a\) should be avoided. |
| n6.11 | \(a^{1/n}\); \(\sqrt[n]{a}\) |
\(a\) to the power \(1/n\), \(n\)th root of \(a\) | \(\sqrt[n]{}a\) should be avoided. |
| n6.12 | \(\bar{x}\); \(\bar{x}_a\) |
arithmetic mean of \(x\) | Other means include: harmonic mean \(\bar{x}_h\); geometric mean \(\bar{x}_g\); quadratic mean / root mean square \(\bar{x}_q\) or \(\bar{x}_{rms}\). \(\bar{x}\) is also used for the complex conjugate of \(x\), see n11.6. |
| n6.13 | \(\operatorname{sgn} a\) | sign function of \(a\) | For real \(a\): \(\operatorname{sgn} a=1\quad (a>0)\); \(\operatorname{sgn} a=-1\quad (a<0)\); \(\operatorname{sgn} 0=0\); See n11.7. |
| n6.14 | \(\inf M\) | infimum of \(M\) | Greatest lower bound of a non-empty set \(M\). |
| n6.15 | \(\sup M\) | supremum of \(M\) | Least upper bound of a non-empty set \(M\). |
| n6.16 | \(\lvert a\rvert\) | absolute value of \(a\) | \(\operatorname{abs} a\) can also be used. |
| n6.17 | \(\lfloor a\rfloor\) | floor greatest integer less than or equal to real \(a\) |
Example: \(\lfloor 2.4\rfloor = 2\); \(\lfloor -2.4\rfloor = -3\). |
| n6.18 | \(\lceil a\rceil\) | ceiling least integer greater than or equal to real \(a\) |
Example: \(\lceil 2.4\rceil = 3\); \(\lceil -2.4\rceil = -2\). |
| n6.19 | \(\min(a, b)\); \(\min\{a, b\}\) |
minimum of \(a\) and \(b\) | Can be extended to finite sets. For the minimum of an infinite set, \(\inf\) is recommended, see n6.14 |
| n6.20 | \(\max(a, b)\); \(\max\{a, b\}\) |
maximum of \(a\) and \(b\) | Can be extended to finite sets. For the maximum of an infinite set, \(\sup\) is recommended, see n6.15 |
| n6.21 | \(n \bmod m\) | remainder of \(n\) modulo \(m\) | For positive integers \(n\), \(m\): \((\exists~q\in\mathbf{N}, r\in[0, m))~~n=qm+r\); where \(r=n \bmod m\). |
| n6.22 | \(\gcd(a, b)\); \(\gcd\{a, b\}\) |
greatest common divisor of integers \(a\) and \(b\) | Can be extended to finite sets. Can be written as \((a, b)\) when no confusion arises. |
| n6.23 | \(\operatorname{lcm}(a, b)\); \(\operatorname{lcm}\{a, b\}\) |
least common multiple of integers \(a\) and \(b\) | Can be extended to finite sets. Can be written as \([a, b]\) when no confusion arises; \((a, b)[a, b]=\lvert ab\rvert\). |
| n6.24 | \([P]\) | Iverson bracket | If proposition \(P\) is true, then \([P]=1\), otherwise \([P]=0\). |
| n6.25 | \(a\uparrow b\)； \(a\uparrow^{n} b\) |
Knuth's up-arrow notation | For non-negative integers \(a,b,n\)： \(a\uparrow^{n} b=a~\underbrace{\uparrow\dots\uparrow}_{n \text{ times}}~b\)； \(a\uparrow^{0} b=ab\)； \(a\uparrow^{1} b=a\uparrow b=a^b\)； \(a\uparrow^{n} 0=1\quad(n>0)\)； \(a\uparrow^{n}b=a\uparrow^{n-1}(a\uparrow^{n}(b-1))\). |
| n6.26 | \([x^n]f(x)\) | coefficient of \(x^n\) in polynomial/formal power series/formal Laurent series \(f(x)\) | If \(\displaystyle f(x)=\sum_{i} a_ix^i\), then \([x^n]f(x)=a_n\)； Can be extended to multivariate cases, e.g., if \(\displaystyle f(x,y)=\sum_{i,j}a_{i,j}x^iy^j\), then \([x^ny^m]f(x,y)=a_{n,m}\). |
Combinatorics
In this section, \(n\) and \(k\) are natural numbers, \(a\) is a complex number, and \(k\leq n\).
Functions
| ID | Symbol, Expression | Meaning, Equivalent Expression | Remarks and Examples |
|---|---|---|---|
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