mastorna.it
IT

Mathematical symbols reference

A searchable reference to symbols commonly found in discrete mathematics, algorithms and theoretical computer science.

Logic and quantifiers

SymbolRead asMeaningExample
βˆƒThere existsAt least one element satisfies the property.βˆƒ x ∈ β„• : x > 5
βˆƒ!There exists a uniqueExactly one element satisfies the property.βˆƒ! x ∈ β„• : x + 1 = 2
βˆ€For everyThe property holds for every element under consideration.βˆ€ n ∈ β„•, n + 1 > n
Β¬NotNegates a proposition.Β¬(x > 5)
∧AndBoth conditions must be true.x > 0 ∧ x < 10
∨OrAt least one condition must be true.x < 0 ∨ x > 10
β‡’ImpliesIf the left-hand proposition is true, so is the right-hand one.x > 2 β‡’ x > 0
⇐Is implied byThe right-hand proposition implies the left-hand one.x > 0 ⇐ x > 2
⇔If and only ifThe two propositions are equivalent.n is even ⇔ n = 2k
:Such thatIntroduces a condition or property.{x ∈ β„• : x > 5}
∴ThereforeIntroduces a conclusion.x = 2 ∴ x² = 4

Relations

SymbolRead asMeaningExample
=Equal toThe two values coincide.x = 5
β‰ Not equal toThe two values do not coincide.x β‰  0
<Less thanThe value on the left is smaller.3 < 5
>Greater thanThe value on the left is larger.8 > 2
≀Less than or equal toLess than or equal to.x ≀ 10
β‰₯Greater than or equal toGreater than or equal to.n β‰₯ nβ‚€
β‰ˆApproximately equal toApproximate equality.Ο€ β‰ˆ 3.14159
≑Equivalent toEquivalence or identity, depending on context.a ≑ b (mod n)
∝Proportional toOne quantity varies in proportion to another.y ∝ x

Sets

SymbolRead asMeaningExample
∈Belongs toThe element belongs to the set.3 ∈ β„•
βˆ‰Does not belong toThe element does not belong to the set.βˆ’2 βˆ‰ β„•
βˆ‹ContainsThe inverse membership relation.β„• βˆ‹ 3
βˆ…Empty setA set containing no elements.A = βˆ…
βŠ†Subset ofEvery element of the first set belongs to the second.{1,2} βŠ† β„•
βŠ‚Proper subset ofA subset that is not equal to the other set.{1,2} βŠ‚ {1,2,3}
βŠ‡Superset ofThe inverse relation of βŠ†.β„• βŠ‡ {1,2}
βˆͺUnionElements belonging to at least one of the two sets.A βˆͺ B
∩IntersectionElements belonging to both sets.A ∩ B
\DifferenceElements in the first set but not in the second.A \ B
|A|Cardinality of AThe number of elements in the set.|{a,b,c}| = 3
A Γ— BCartesian productThe set of ordered pairs formed from elements of A and B.(a,b) ∈ A Γ— B

Number sets

SymbolNameContentsExample
β„•Natural numbers0, 1, 2, 3, … depending on the chosen convention.5 ∈ β„•
β„€Integers…, βˆ’2, βˆ’1, 0, 1, 2, β€¦βˆ’3 ∈ β„€
β„šRational numbersNumbers representable as a fraction of integers.1/3 ∈ β„š
ℝReal numbersIncludes rational and irrational numbers.√2 ∈ ℝ
β„‚Complex numbersNumbers of the form a + bi.2 + 3i ∈ β„‚
Inclusion relation: β„• βŠ† β„€ βŠ† β„š βŠ† ℝ βŠ† β„‚.

Intervals

NotationMeaningEquivalent formNotes
[a,b]Both endpoints included.a ≀ x ≀ bClosed interval.
(a,b)Both endpoints excluded.a < x < bOpen interval; ]a,b[ is another convention.
[a,b)a included, b excluded.a ≀ x < bHalf-open interval.
(a,b]a excluded, b included.a < x ≀ bHalf-open interval.
(βˆ’βˆž,a)All values less than a.x < aAn infinite endpoint is always open.
[a,+∞)All values greater than or equal to a.x β‰₯ aAn infinite endpoint is always open.

Functions

SymbolRead asMeaningExample
f: A β†’ Bf from A to BThe function has domain A and codomain B.f: β„• β†’ β„•
x ↦ f(x)x maps to f(x)Describes the transformation applied to the argument.x ↦ xΒ²
f(x)f of xThe value produced by the function at x.f(3) = 9
f⁻¹Inverse functionReverses the function when an inverse exists.f⁻¹(f(x)) = x
f ∘ gf composed with gApply g first, then f.(f ∘ g)(x) = f(g(x))
dom(f)Domain of fThe set of allowed input values.dom(f) = ℝ
Im(f)Image of fThe set of values actually produced.Im(f) βŠ† B

Operators, sequences and rounding

SymbolRead asMeaningExample
Ξ£SummationThe sum of a sequence of terms.Ξ£α΅’β‚Œβ‚βΏ i
Ξ ProductThe product of a sequence of terms.Ξ α΅’β‚Œβ‚βΏ i = n!
|x|Absolute valueThe distance of x from zero.|βˆ’5| = 5
⌊xβŒ‹FloorThe greatest integer less than or equal to x.⌊3.8βŒ‹ = 3
⌈xβŒ‰CeilingThe least integer greater than or equal to x.⌈3.2βŒ‰ = 4
n!FactorialThe product of positive integers from 1 to n.5! = 120
∞InfinityUnbounded growth or extent.n β†’ ∞
log nLogarithm of nOften base 2 in algorithms unless otherwise stated.logβ‚‚ 8 = 3

Asymptotic notation

SymbolRead asIntuitive meaningExample
O(g(n))Big O of g(n)Upper bound: f grows no faster than g.3n² + 2 ∈ O(n²)
Ω(g(n))Big Omega of g(n)Lower bound: f grows at least as fast as g.3n² + 2 ∈ Ω(n²)
Θ(g(n))Theta of g(n)Tight bound: f grows like g up to constant factors.3n² + 2 ∈ Θ(n²)
o(g(n))Little o of g(n)f grows strictly more slowly than g.n ∈ o(n²)
Ο‰(g(n))Little omega of g(n)f grows strictly faster than g.nΒ² ∈ Ο‰(n)

MathML example: definition of Θ

This formula is written directly in MathML without external libraries.

Θ ( g (n) ) = { f (n) : βˆƒ c1 >0 , βˆƒ c2 >0 , βˆƒ n0 >0 : 0 ≀ c1 g(n) ≀ f(n) ≀ c2 g(n) , βˆ€ n β‰₯ n0 }
Read as: there are positive constants c₁, cβ‚‚ and nβ‚€ such that, for every n β‰₯ nβ‚€, f(n) lies between c₁g(n) and cβ‚‚g(n).

Standalone page: no framework, CDN or external JavaScript required.