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. (# or ♯ may 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 | A ⇒ B 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 A ∧ B 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) u ∧ v 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 A ∨ B 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 A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. | (¬A) ⊕ A is always true, A ⊕ A 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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {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 use ≡ to 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 |