Help:Mathematical symbols and expressions

This is a list of mathematical symbols and expressions used on.
 * For a glossary of Asphalt-specific terms, see Asphalt Wiki:Glossary.
 * For a general FANDOM glossary, see Help:Glossary.

As the list is meant for players not familiar with a certain symbol or expression, the following table is sorted by symbols, not by their meanings.
 * Symbols are sorted under !$@. They may sometimes contain additional grey variables for clarity purposes, like $${\color{grey}n}!$$ or $$\{{\color{grey}a}, {\color{grey}b}\}$$.
 * Symbols that resemble letters are listed both under !$@ and the corresponding letter.
 * Expressions that are commonly used with specific letters are sorted under the first letter that appears in them. For example, a player searching for the meaning of $$\binom{n}{k}$$ will find it under N, and also under !$@ because its main elements are brackets, but not under B for "binomial coefficient".
 * Greek letters can be found under !$@ as well as under their Latin equivalents.

{| class="wikitable" !colspan="3" style="background:;"|

!$@
!colspan="3" style="background:#1c1a1a;"| Basic symbols !colspan="3" style="background:#1c1a1a;"| Symbols pointing in a direction !colspan="3" style="background:#1c1a1a;"| Brackets !colspan="3" style="background:#1c1a1a;"| Bars !colspan="3" style="background:#1c1a1a;"| Letter-based symbols !colspan="3" style="background:;"|
 * $${\color{grey}a}\cdot{\color{grey}b}$$
 * Multiplication
 * $$a \cdot b$$ read: "$$a$$ times $$b$$" (preferred to $$\times$$ in mathematical contexts on this wiki to avoid confusion with the letter x)
 * $${\color{grey}n}!$$
 * Factorial of $$n$$
 * $$5! = 1 \cdot  2  \cdot  3  \cdot  4  \cdot  5$$
 * $$\infty$$
 * Infinity
 * $$\infty$$
 * Infinity
 * Infinity
 * Less than Greater than
 * $$a < b$$ read: "$$a$$ is less than $$b$$." ($$a$$ is not equal to $$b$$.)
 * $$\subset$$ $$\supset$$
 * Proper (strict) subset of Proper (strict) superset of
 * $$A \subset B$$ read: "$$A$$ is a proper subset of $$B$$." ($$A$$ is not equal to $$B$$.)
 * $$\le$$ $$\ge$$
 * Less than or equal to Greater than or equal to
 * $$a \le b$$ read: "$$a$$ is less than or equal to $$b$$."
 * $$\subseteq$$ $$\supseteq$$
 * Subset of or equal to Superset of or equal to
 * $$A \subseteq B$$ read: "$$A$$ is a subset of or equal to $$B$$."
 * $$\land$$
 * And (for logical statements and equations)
 * $$A \cap B = \{ x \mid x \in A \land x \in B\}$$ read: "$$A$$ intersected with $$B$$ equals the set of all $$x$$ such that $$x \in A$$ and $$x \in B$$."
 * $$\cap$$
 * Intersection (for sets)
 * $$A \cap B = \{ x \mid x \in A \land x \in B\}$$ read: "$$A$$ intersected with $$B$$ equals the set of all $$x$$ such that $$x \in A$$ and $$x \in B$$."
 * $$\lor$$
 * Or (for logical statements and equations)
 * $$x^2 = 9 \iff x = 3 \lor x = -3$$ read: "$$x^2$$ equals $$9$$ if and only if $$x = 3$$ or $$x = -3$$."
 * $$\cup$$
 * Union (for sets)
 * $$A \cup B = \{ x \mid x \in A \lor x \in B\}$$ read: "The union of $$A$$ and $$B$$ equals the set of all $$x$$ such that $$x \in A$$ or $$x \in B$$."
 * $$\to$$
 * Approaches
 * $$\overline{X}_n \, \to \, \mu \ \ \mathrm{when}\ \ n \to \infty$$ read: "$$X_n$$ bar approaches $$\mu$$ when $$n$$ approaches infinity."
 * $$\Rightarrow$$
 * Implication (for logical statements and equations)
 * $$x = 3 \Rightarrow x^2 = 9$$ read: "implies" (It follows from $$x = 3$$ that $$x^2 = 9$$, but it doesn't follow from $$x^2 = 9$$ that $$x = 3$$ since $$x$$ could also be $$-3$$.)
 * $$\iff$$
 * Equivalence (for logical statements and equations)
 * $$2x = 10 \iff x = 5$$ read: "if and only if" (Both statements imply each other.)
 * $$\to$$
 * Approaches
 * $$\overline{X}_n \, \to \, \mu \ \ \mathrm{when}\ \ n \to \infty$$ read: "$$X_n$$ bar approaches $$\mu$$ when $$n$$ approaches infinity."
 * $$\Rightarrow$$
 * Implication (for logical statements and equations)
 * $$x = 3 \Rightarrow x^2 = 9$$ read: "implies" (It follows from $$x = 3$$ that $$x^2 = 9$$, but it doesn't follow from $$x^2 = 9$$ that $$x = 3$$ since $$x$$ could also be $$-3$$.)
 * $$\iff$$
 * Equivalence (for logical statements and equations)
 * $$2x = 10 \iff x = 5$$ read: "if and only if" (Both statements imply each other.)
 * $$\iff$$
 * Equivalence (for logical statements and equations)
 * $$2x = 10 \iff x = 5$$ read: "if and only if" (Both statements imply each other.)
 * $$({\color{grey}a}, {\color{grey}b})$$
 * Tuple (An $$n$$-tuple is an ordered list of $$n$$ elements. $$(a, b)$$ is not the same as $$(b, a)$$.)
 * $$(a, b, c)$$ read: "the 3-tuple (triple) of $$a, b, c$$"
 * $$\binom$$
 * $$[{\color{grey}a}, {\color{grey}b}]$$
 * Closed interval from $$a$$ to $$b$$ (including $$a$$ and $$b$$)
 * $$\mathopen{[}a,b\mathclose{]} = \{x \in \R \mid a \le x \le b \}$$
 * $$]{\color{grey}a}, {\color{grey}b}[$$
 * Open interval from $$a$$ to $$b$$ (excluding $$a$$ and $$b$$)
 * $$\mathopen{]}a,b\mathclose{[} = \{x \in \R \mid a < x < b \}$$
 * $$\{\}$$
 * $$\{{\color{grey}a}, {\color{grey}b}\}$$
 * Set (unordered, $$\{a, b\}$$ is the same as $$\{b, a\}$$)
 * $$\{a, b, c\}$$ read: "the set of $$a$$, $$b$$, and $$c$$"
 * $$\{ {\color{grey}x} \mid {\color{grey}A(x)}\}$$
 * Set-builder notation (the set of all $$x$$ for which $$A(x)$$ is true)
 * $$\{x \in \N \mid x < 5 \} = \{1, 2, 3, 4\}$$ read: "the set of all $$x$$ that are elements of $$\N$$ such that $$x < 5$$"
 * $$\{\}$$
 * $$\{{\color{grey}a}, {\color{grey}b}\}$$
 * Set (unordered, $$\{a, b\}$$ is the same as $$\{b, a\}$$)
 * $$\{a, b, c\}$$ read: "the set of $$a$$, $$b$$, and $$c$$"
 * $$\{ {\color{grey}x} \mid {\color{grey}A(x)}\}$$
 * Set-builder notation (the set of all $$x$$ for which $$A(x)$$ is true)
 * $$\{x \in \N \mid x < 5 \} = \{1, 2, 3, 4\}$$ read: "the set of all $$x$$ that are elements of $$\N$$ such that $$x < 5$$"
 * $$\{ {\color{grey}x} \mid {\color{grey}A(x)}\}$$
 * Set-builder notation (the set of all $$x$$ for which $$A(x)$$ is true)
 * $$\{x \in \N \mid x < 5 \} = \{1, 2, 3, 4\}$$ read: "the set of all $$x$$ that are elements of $$\N$$ such that $$x < 5$$"
 * $$\{x \in \N \mid x < 5 \} = \{1, 2, 3, 4\}$$ read: "the set of all $$x$$ that are elements of $$\N$$ such that $$x < 5$$"
 * $${\color{grey}1.}\bar{\color{grey}{3}}$$
 * Repeating decimal
 * $$1.\bar3 = 1.3333\dots$$ read: "one point three repeating"
 * $$\bar{\color{grey}{x}}$$
 * $$\bar{\color{grey}A}$$
 * Complementary event
 * read: "$$A$$ bar"
 * $${\color{grey}\mathrm{P}(A} \mid {\color{grey}B)}$$
 * $${\color{grey}\{x} \mid {\color{grey}A(x)\}}$$
 * $$|{\color{grey}a}|$$
 * Absolute value of $$a$$ (for numbers, the non-negative value of $$a$$ without regard to its sign)
 * $$|-2| = 2$$
 * $$|{\color{grey}A}|$$
 * Cardinality of $$A$$ (for sets, the number of elements in the set $$A$$)
 * $$|A| = 4$$
 * $$\lfloor {\color{grey}a} \rfloor$$
 * Floor function (for numbers, the greatest integer less than or equal to $$a$$)
 * $$\lfloor 2.8 \rfloor = 2$$ read: "floor of 2.8"
 * $$\lceil {\color{grey}a} \rceil$$
 * Ceiling function (for numbers, the least integer greater than or equal to $$a$$)
 * $$\lceil 2.2 \rceil = 3$$ read: "ceiling of 2.2"
 * $$|{\color{grey}a}|$$
 * Absolute value of $$a$$ (for numbers, the non-negative value of $$a$$ without regard to its sign)
 * $$|-2| = 2$$
 * $$|{\color{grey}A}|$$
 * Cardinality of $$A$$ (for sets, the number of elements in the set $$A$$)
 * $$|A| = 4$$
 * $$\lfloor {\color{grey}a} \rfloor$$
 * Floor function (for numbers, the greatest integer less than or equal to $$a$$)
 * $$\lfloor 2.8 \rfloor = 2$$ read: "floor of 2.8"
 * $$\lceil {\color{grey}a} \rceil$$
 * Ceiling function (for numbers, the least integer greater than or equal to $$a$$)
 * $$\lceil 2.2 \rceil = 3$$ read: "ceiling of 2.2"
 * $$\lceil {\color{grey}a} \rceil$$
 * Ceiling function (for numbers, the least integer greater than or equal to $$a$$)
 * $$\lceil 2.2 \rceil = 3$$ read: "ceiling of 2.2"
 * $$\forall$$
 * $$\in$$
 * $$\varnothing$$
 * $$\varepsilon$$
 * $$\mu$$
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\varepsilon$$
 * $$\mu$$
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$

A
!colspan="3" style="background:;"|
 * $$\land$$
 * $$\forall$$
 * For all
 * $$x+x=2x \; \forall x \in \R$$ read: "$$x+x=2x$$ for all $$x$$ in $$\R$$."
 * $$\mathrm{a. s.}$$
 * Almost surely
 * $$\xrightarrow{\mathrm{a. s.}}$$
 * Almost sure convergence
 * $${X_n \, \xrightarrow{\mathrm{a. s.}} \, X}$$ read: "$X_n$ converges almost surely towards $X$."
 * $$\mathrm{a. s.}$$
 * Almost surely
 * $$\xrightarrow{\mathrm{a. s.}}$$
 * Almost sure convergence
 * $${X_n \, \xrightarrow{\mathrm{a. s.}} \, X}$$ read: "$X_n$ converges almost surely towards $X$."
 * Almost sure convergence
 * $${X_n \, \xrightarrow{\mathrm{a. s.}} \, X}$$ read: "$X_n$ converges almost surely towards $X$."

E
!colspan="3" style="background:;"|
 * $$e$$
 * The mathematical constant $$e$$ ($$\approx 2.71828$$)
 * $$e = \displaystyle\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$
 * $$\operatorname{E}[X]$$
 * Expected value
 * read: "Expected value of $$X$$"
 * $$\varepsilon$$
 * Epsilon (Greek, lower case): often used to denote an arbitrarily small positive quantity, particularly for the definition of a
 * $$\in$$
 * Element of
 * $$x \in \R$$ read: "$$x$$ is an element of $$\R$$."
 * $$\Sigma$$
 * $$\in$$
 * Element of
 * $$x \in \R$$ read: "$$x$$ is an element of $$\R$$."
 * $$\Sigma$$
 * $$x \in \R$$ read: "$$x$$ is an element of $$\R$$."
 * $$\Sigma$$
 * $$\Sigma$$

I
!colspan="3" style="background:;"|
 * $$\mathrm{i. i. d.}$$
 * Independent and identically distributed
 * Independent and identically distributed

L
!colspan="3" style="background:;"|
 * $$\mathrm{lim}$$
 * Limit (a value that a sequence or function "tends to")
 * $$\displaystyle\lim_{n\to\infty}0.3^n = 0$$ read: "the limit of $$0.3^n$$, as $$n$$ approaches infinity"
 * $$\ln x$$
 * Natural logarithm (The natural logarithm of $$x$$ is the power to which would have to be raised to equal $$x$$.)
 * $$\ln e^x = x$$
 * Natural logarithm (The natural logarithm of $$x$$ is the power to which would have to be raised to equal $$x$$.)
 * $$\ln e^x = x$$

M
!colspan="3" style="background:;"|
 * $$\mu$$
 * Mu (Greek, lower case): population mean
 * read: "mew" [mjuː]
 * read: "mew" [mjuː]

N
!colspan="3" style="background:;"|
 * $$\N$$
 * The set of natural numbers
 * $$\binom{n}{k}$$
 * Binomial coefficient
 * $$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$ read: "n choose k"
 * $$\binom{n}{k}$$
 * Binomial coefficient
 * $$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$ read: "n choose k"

O
!colspan="3" style="background:;"|
 * $$\varnothing$$
 * Empty set
 * $$\varnothing = \{\}$$ (The symbol was actually derived from the letter Ø in the Norwegian alphabet. )
 * $$\omega$$
 * Omega (Greek, lower case): element of a sample space
 * $$\Omega$$
 * Omega (Greek, upper case): sample space
 * $$\Omega$$
 * Omega (Greek, upper case): sample space
 * $$\Omega$$
 * Omega (Greek, upper case): sample space

P
!colspan="3" style="background:;"|
 * $$\mathrm P(X)$$
 * Probability
 * read: "the probability of $$X$$"
 * $$\mathrm{P}(A \mid B)$$
 * Conditional probability
 * read: "the conditional probability of $$A$$ given $$B$$" or: "the probability of $$A$$ under the condition $$B$$"
 * $$\xrightarrow{\mathrm P}$$
 * Convergence in probability
 * $${X_n \, \xrightarrow{\mathrm P} \, X}$$ read: "$X_n$ converges in probability towards $X$."
 * $$\Pi$$
 * Pi (Greek, upper case): product
 * $$\prod_{i=1}^4 a_i = a_1\cdot a_2\cdot a_3\cdot a_4$$ read: "product over $$i$$ from 1 to 4 of $$a_i$$" or: "product, as $$i$$ goes from 1 to 4, of $$a_i$$"
 * $$\Pi$$
 * Pi (Greek, upper case): product
 * $$\prod_{i=1}^4 a_i = a_1\cdot a_2\cdot a_3\cdot a_4$$ read: "product over $$i$$ from 1 to 4 of $$a_i$$" or: "product, as $$i$$ goes from 1 to 4, of $$a_i$$"
 * $$\prod_{i=1}^4 a_i = a_1\cdot a_2\cdot a_3\cdot a_4$$ read: "product over $$i$$ from 1 to 4 of $$a_i$$" or: "product, as $$i$$ goes from 1 to 4, of $$a_i$$"

Q
!colspan="3" style="background:;"|
 * $$\mathbb{Q}$$
 * The set of rational numbers
 * The set of rational numbers

R
!colspan="3" style="background:;"|
 * $$\R$$
 * The set of real numbers
 * The set of real numbers

S
!colspan="3" style="background:;"|
 * $$s$$
 * Standard deviation of a sample
 * $$s^2$$
 * Variance of a sample
 * $$\sigma$$
 * Sigma (Greek, lower case): standard deviation of a population
 * $$\sigma^2$$
 * Sigma (Greek, lower case, squared): variance of a population
 * $$\Sigma$$
 * Sigma (Greek, upper case): sum
 * $$\displaystyle\sum_{i=1}^{4} a_i = a_1 + a_2 + a_3 + a_4$$ read: "sum over $$i$$ from 1 to 4 of $$a_i$$" or: "sum, as $$i$$ goes from 1 to 4, of $$a_i$$"
 * $$\sigma^2$$
 * Sigma (Greek, lower case, squared): variance of a population
 * $$\Sigma$$
 * Sigma (Greek, upper case): sum
 * $$\displaystyle\sum_{i=1}^{4} a_i = a_1 + a_2 + a_3 + a_4$$ read: "sum over $$i$$ from 1 to 4 of $$a_i$$" or: "sum, as $$i$$ goes from 1 to 4, of $$a_i$$"
 * $$\Sigma$$
 * Sigma (Greek, upper case): sum
 * $$\displaystyle\sum_{i=1}^{4} a_i = a_1 + a_2 + a_3 + a_4$$ read: "sum over $$i$$ from 1 to 4 of $$a_i$$" or: "sum, as $$i$$ goes from 1 to 4, of $$a_i$$"
 * $$\displaystyle\sum_{i=1}^{4} a_i = a_1 + a_2 + a_3 + a_4$$ read: "sum over $$i$$ from 1 to 4 of $$a_i$$" or: "sum, as $$i$$ goes from 1 to 4, of $$a_i$$"

U
!colspan="3" style="background:;"|
 * $$\cup$$
 * $$\cap$$
 * $$\cap$$
 * $$\cap$$
 * $$\cap$$
 * $$\cap$$

V
!colspan="3" style="background:;"|
 * $$\lor$$
 * $$\forall$$
 * $$\land$$
 * $$\forall$$
 * $$\land$$
 * $$\forall$$
 * $$\land$$
 * $$\land$$
 * $$\land$$
 * $$\land$$

W
!colspan="3" style="background:;"|
 * $$\omega$$
 * $$\omega$$

X
!colspan="3" style="background:;"|
 * $$\bar{x}$$
 * Arithmetic mean, sample mean
 * read: "$$x$$ bar"
 * read: "$$x$$ bar"

Z

 * $$\Z$$
 * The set of integers
 * }
 * }
 * }