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This is a list of mathematical symbols and expressions used on Asphalt Wiki.

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 reasons, 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.

 !$@

Basic symbols
$ a^b $ Exponentiation
(for numbers)
$ a^b = \underbrace{a \cdot a \cdot \ldots \cdot a}_{b\, \mathrm{times}} $
read: "$ a $ raised to the power of $ b $"
or most briefly: "$ a $ to the $ b $"
$ 2^A $ Alternative notation for the →power set of $ A $
(for sets)
$ {\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 $
$ \sim $ Has the probability distribution
(for random variables)
$ X \sim D $
read: "$ X $ has (the probability) distribution $ D $."
or: "$ X $ is $ D $-distributed."
$ \approx $ Approximately equal to
(for numbers)
$ \frac23 \approx 0.67 $
read: "Two thirds is approximately equal to 0.67."
$ \infty $ Infinity
Symbols pointing in a direction
$ < $
$ > $
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."
$ \xrightarrow{{\color{grey}\mathrm{a.\ s.}}} $ Almost sure convergence
$ \xrightarrow{{\color{grey}\mathrm P}} $ Convergence in probability
$ \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.)
Brackets
$ ({\color{grey}a}, {\color{grey}b}) $ 1. Tuple
(An $ n $-tuple is an ordered list of $ n $ elements. $ (a, b) $ is not the same as $ (b, a) $.)
2. Alternative notation for the →open interval $ \mathopen{]}a,b\mathclose{[} $
$ (a, b, c) $
read: "the 3-tuple (triple) of $ a, b, c $"
$ \binom{{\color{grey}n}}{{\color{grey}k}} $ Binomial coefficient
$ [{\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 \} $
(preferred to $ (a, b) $ on this wiki to avoid confusion with the →tuple $ (a, b) $)
$ \{\} $ Empty set
$ \{{\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 $"
Bars
$ {\color{grey}1.}\bar{\color{grey}{3}} $ Repeating decimal $ 1.\bar3 = 1.3333\dots $
read: "one point three repeating"
$ \bar{\color{grey}{x}} $ Arithmetic mean
$ \bar{\color{grey}A} $ Complementary event read: "$ A $ bar"
$ {\color{grey}\mathrm{P}(A} \mid {\color{grey}B)} $ Conditional probability
$ {\color{grey}\{x} \mid {\color{grey}A(x)\}} $ Set-builder notation
$ |{\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"
Letter-based symbols
$ \forall $ For all
$ \in $ Element of
$ \varnothing $ Empty set
$ \varepsilon $ Epsilon (lower case)
$ \mu $ Mu (lower case)
$ \omega $ Omega (lower case)
$ \Omega $ Omega (upper case)
$ \Pi $ Pi (upper case)
$ \sigma $ Sigma (lower case)
$ \sigma^2 $ Sigma² (lower case)
$ \Sigma $ Sigma (upper case)

A

$ \land $ And
$ \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 $."

B

$ \mathrm{B}(n, p) $ Binomial distribution $ X \sim \mathrm{B}(n, p) $
read: "$ X $ is binomially distributed with parameters $ n $ and $ p $."
$ \mathrm{Ber}(p) $ Bernoulli distribution, also written as $ \mathrm{Bernoulli}(p) $ $ X \sim \mathrm{Ber}(p) $
read: "$ X $ is Bernoulli distributed with parameter $ p $."

E

$ 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 →limit
$ \in $ Element of $ x \in \R $
read: "$ x $ is an element of $ \R $."
$ \Sigma $ Sigma (upper case)

I

$ \mathrm{i. i. d.} $ Independent and identically distributed

L

$ \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 →e would have to be raised to equal $ x $.)
$ \ln e^x = x $

M

$ \mu $ Mu (Greek, lower case):
population mean, also used for →expected value
read: "mew" [mjuː]

N

$ \N $ The set of natural numbers
$ \binom{n}{k} $ Binomial coefficient $ \binom{n}{k} = \frac{n!}{k! (n-k)!} $
read: "n choose k"

O

$ \varnothing $ Empty set $ \varnothing = \{\} $
(The symbol was actually derived from the letter Ø in the Norwegian alphabet.[1])
$ \omega $ Omega (Greek, lower case): element of a sample space
$ \Omega $ Omega (Greek, upper case): sample space

P

$ \mathrm P(A) $ Probability
(for events)
read: "the probability of $ A $"
$ \mathcal P(A) $ Power set
(for sets, the set of all →subsets of $ A $, including the →empty set and $ A $ itself)
$ \mathcal P(A) = 2^A $
read: "the power set of $ A $"
  • $ \mathcal P $ is written in calligraphy typeface to avoid confusion with the →probability $ \mathrm P(A) $.
  • If $ |A| = n $, then $ |\mathcal P(A)| = 2^n $, which is the motivation for the alternative notation $ 2^A $.
$ \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 $"

Q

$ \mathbb{Q} $ The set of rational numbers

R

$ \R $ The set of real numbers

S

$ 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 $"

U

$ \cup $ Union
$ \cap $ Intersection

V

$ \lor $ Or
$ \forall $ For all
$ \land $ And
$ \mathrm{Var}(X) $ Variance of $ X $, usually denotes the population variance $ \sigma^2 $

W

$ \omega $ Omega (lower case)

X

$ \bar{x} $ Arithmetic mean, sample mean read: "$ x $ bar"

Z

$ \Z $ The set of integers

References

  1. Weill, André (1992). The Apprenticeship of a Mathematician. Birkhäuser Verlag. p. 114. Retrieved on 2019-07-21.
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