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User:Alksentrs/Table of mathematical symbols (testing)

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Test version of the article Table of mathematical symbols.

Symbol
(HTML)
Symbol
(TeX)
Name Explanation Examples
Read as
Category
|…|
absolute value or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5| = 5

i | = 1

| 3 + 4i | = 5
absolute value (modulus) of
numbers
Euclidean distance |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
geometry
determinant |A| means the determinant of the matrix A
determinant of
matrix theory
cardinality |X| means the cardinality of the set X.

(# ormay be used instead as described below.)
|{3, 5, 7, 9}| = 4.
cardinality of; size of
set theory
#



cardinality #X means the cardinality of the set X.

(|…| may be used instead as described above.)
#{4, 6, 8} = 3
cardinality of; size of
set theory
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
number theory
conditional probability A single vertical bar is used to describe the probability of an event given another event happening.
P(A|B) means a given b.
If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4
given
probability
!
factorial n! is the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
T

tr


transpose Swap rows for columns If then .
transpose
matrix operations
~
probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics
row equivalence A~B means that B can be generated by using a series of elementary row operations on A
is row equivalent to
matrix theory
same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar; poorly approximates
approximation theory
asymptotically equivalent f ~ g means . x ~ x+1

is asymptotically equivalent to
asymptotic analysis
equivalence relation a ~ b means (and equivalently ). 1 ~ 5 mod 4

are in the same equivalence class
everywhere
approximately equal x ≈ y means x is approximately equal to y. π ≈ 3.14159
is approximately equal to
everywhere
isomorphism G ≈ H means that group G is isomorphic (structurally identical) to group H.

(≅ can also be used for isomorphic, as described below.)
Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
normal subgroup N ◅ G means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
ideal I ◅ R means that I is an ideal of ring R. (2) ◅ Z
is an ideal of
ring theory
therefore Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
therefore; so; hence
everywhere
because Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one.
because; since
everywhere








material implication AB means if A is true then B is also true; if A is false then nothing is said about B.

(→ may mean the same as, or it may have the meaning for functions given below.)

(⊃ may mean the same as, or it may have the meaning for superset given below.)
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if … then
propositional logic, Heyting algebra




material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜


logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).

(Old notation) uv means the cross product of vectors u and v.
n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and; min
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or; max
propositional logic, lattice theory




exclusive or The statement AB is true when either A or B, but not both, are true. AB means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic). Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = {0})
direct sum of
abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even.
there exists
predicate logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n.
there exists exactly one
predicate logic
:=



:⇔








definition x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
is defined as; equal by definition
everywhere
congruence △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
geometry
isomorphic G ≅ H means that group G is isomorphic (structurally identical) to group H.

(≈ can also be used for isomorphic, as described above.)
.
is isomorphic to
abstract algebra