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 their meanings.
 * Symbols are sorted under !$@. They may sometimes contain variable names for demonstration purposes, like $$n!$$ or $$[a, b]$$.
 * 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:;"|
 * 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
 * $$\cap$$
 * Intersection
 * $$\lor$$
 * Or
 * $$\cup$$
 * Union
 * $$\in$$
 * $$\infty$$
 * Infinity
 * $$\forall$$
 * $$\varnothing$$
 * $$\to$$
 * Approaches
 * $$n!$$
 * Factorial of $$n$$
 * $$5! = 1 \cdot  2  \cdot  3  \cdot  4  \cdot  5$$
 * $$\binom{n}{k}$$
 * $$[a, b]$$
 * Closed interval from $$a$$ to $$b$$, including $$a$$ and $$b$$
 * $$\mathopen{[}a,b\mathclose{]} = \{x \in \R \mid a \le x \le b \}$$
 * $$]a, b[$$
 * Open interval from $$a$$ to $$b$$, excluding $$a$$ and $$b$$
 * $$\mathopen{]}a,b\mathclose{[} = \{x \in \R \mid a < x < b \}$$
 * $$\{\}$$
 * $$|a|$$
 * Absolute value of $$a$$ (for numbers, the non-negative value of $$a$$ without regard to its sign)
 * $$|-2| = 2$$
 * $$|A|$$
 * Cardinality of $$A$$ (for sets, the number of elements in the set $$A$$)
 * $$|A| = 4$$
 * $$\lfloor a \rfloor$$
 * Floor function (for numbers, the greatest integer less than or equal to $$a$$)
 * $$\lfloor 2.8 \rfloor = 2$$ read: "floor of 2.8"
 * $$1.\bar3$$
 * Repeating decimal
 * $$1.\bar3 = 1.3333\dots$$ read: "one point three repeating"
 * $$\bar x$$
 * $$\bar A$$
 * Complementary event
 * read: "$$A$$ bar"
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$[a, b]$$
 * Closed interval from $$a$$ to $$b$$, including $$a$$ and $$b$$
 * $$\mathopen{[}a,b\mathclose{]} = \{x \in \R \mid a \le x \le b \}$$
 * $$]a, b[$$
 * Open interval from $$a$$ to $$b$$, excluding $$a$$ and $$b$$
 * $$\mathopen{]}a,b\mathclose{[} = \{x \in \R \mid a < x < b \}$$
 * $$\{\}$$
 * $$|a|$$
 * Absolute value of $$a$$ (for numbers, the non-negative value of $$a$$ without regard to its sign)
 * $$|-2| = 2$$
 * $$|A|$$
 * Cardinality of $$A$$ (for sets, the number of elements in the set $$A$$)
 * $$|A| = 4$$
 * $$\lfloor a \rfloor$$
 * Floor function (for numbers, the greatest integer less than or equal to $$a$$)
 * $$\lfloor 2.8 \rfloor = 2$$ read: "floor of 2.8"
 * $$1.\bar3$$
 * Repeating decimal
 * $$1.\bar3 = 1.3333\dots$$ read: "one point three repeating"
 * $$\bar x$$
 * $$\bar A$$
 * Complementary event
 * read: "$$A$$ bar"
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\bar x$$
 * $$\bar A$$
 * Complementary event
 * read: "$$A$$ bar"
 * $$\omega$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\Omega$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\Pi$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\sigma^2$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$
 * $$\Sigma$$

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:;"|
 * $$\operatorname{E}$$
 * Expected value
 * $$\operatorname{E}[X]$$ read: "Expected value of $$X$$"
 * $$\in$$
 * Element of
 * $$x \in \R$$ read: "$$x$$ is an element of the set of real numbers."
 * Element of
 * $$x \in \R$$ read: "$$x$$ is an element of the set of real numbers."

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"
 * $$\displaystyle\lim_{n\to\infty}0.3^n = 0$$ read: "the limit of $$0.3^n$$, as $$n$$ approaches infinity"

N
!colspan="3" style="background:;"|
 * $$\binom{n}{k}$$
 * Binomial coefficient
 * $$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$ read: "n choose k"
 * $$\binom{n}{k} = \frac{n!}{k! (n-k)!}$$ 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:;"|
 * $$P$$
 * Probability
 * $$P(X)$$ read: "Probability of $$X$$"
 * $$\xrightarrow{P}$$
 * Convergence in probability
 * $${X_n \, \xrightarrow{P} \, X}$$ 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 $$a_i$$" "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 $$a_i$$" "product, as $$i$$ goes from 1 to 4, of $$a_i$$"

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 $$a_i$$" "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 $$a_i$$" "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 $$a_i$$" "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 $$a_i$$" "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:;"|
 * $$\lor$$
 * $$\forall$$
 * $$\land$$
 * $$\forall$$
 * $$\land$$
 * $$\forall$$
 * $$\land$$
 * $$\land$$
 * $$\land$$
 * $$\land$$

X

 * $$\bar{x}$$
 * Arithmetic mean
 * read: "$$x$$ bar"
 * }
 * }