Главная страница Случайная страница
Разделы сайта
АвтомобилиАстрономияБиологияГеографияДом и садДругие языкиДругоеИнформатикаИсторияКультураЛитератураЛогикаМатематикаМедицинаМеталлургияМеханикаОбразованиеОхрана трудаПедагогикаПолитикаПравоПсихологияРелигияРиторикаСоциологияСпортСтроительствоТехнологияТуризмФизикаФилософияФинансыХимияЧерчениеЭкологияЭкономикаЭлектроника
Table of mathematical symbols
⇐ ПредыдущаяСтр 35 из 35
|
|
| Symbol | Name | Explanation | Examples | ||||||
| Read as | |||||||||
| Category | |||||||||
| = | equality | x = y means x and y represent the same thing or value. | 1 + 1 = 2 | ||||||
| is equal to; equals | |||||||||
| everywhere | |||||||||
| ≠ < >! = | inequation | x ≠ y means that x and y do not represent the same thing or value. (The symbols! = and < > are primarily from computer science. They are avoided in mathematical texts.) | 1 ≠ 2 | ||||||
| is not equal to; does not equal | |||||||||
| everywhere | |||||||||
| < > ≪ ≫ | strict inequality | x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y. | 3 < 4 5 > 4. 0.003 ≪ 1000000 | ||||||
| is less than, is greater than, is much less than, is much greater than | |||||||||
| order theory | |||||||||
| ≤ ≥ | inequality | x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. | 3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 | ||||||
| is less than or equal to, is greater than or equal to | |||||||||
| order theory | |||||||||
| ∝ | proportionality | y ∝ x means that y = kx for some constant k. | if y = 2 x, then y ∝ x | ||||||
| is proportional to | |||||||||
| everywhere | |||||||||
| + | addition | 4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 | ||||||
| plus | |||||||||
| arithmetic | |||||||||
| disjoint union | A 1 + A 2 means the disjoint union of sets A 1 and A 2. | A 1 = {1, 2, 3, 4} ∧ A 2 = {2, 4, 5, 7} ⇒ A 1 + A 2 = {(1, 1), (2, 1), (3, 1), (4, 1), (2, 2), (4, 2), (5, 2), (7, 2)} | |||||||
| the disjoint union of... and... | |||||||||
| set theory | |||||||||
| − | subtraction | 9 − 4 means the subtraction of 4 from 9. | 8 − 3 = 5 | ||||||
| minus | |||||||||
| arithmetic | |||||||||
| negative sign | − 3 means the negative of the number 3. | − (− 5) = 5 | |||||||
| negative; minus | |||||||||
| arithmetic | |||||||||
| set-theoretic complement | A − B means the set that contains all the elements of A that are not in B. | {1, 2, 4} − {1, 3, 4} = {2} | |||||||
| minus; without | |||||||||
| set theory | |||||||||
| × | multiplication | 3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 | ||||||
| times | |||||||||
| arithmetic | |||||||||
| Cartesian product | X × Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. | {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)} | |||||||
| the Cartesian product of... and...; the direct product of... and... | |||||||||
| set theory | |||||||||
| cross product | u × v means the cross product of vectors u and v | (1, 2, 5) × (3, 4, − 1) = (− 22, 16, − 2) | |||||||
| cross | |||||||||
| vector algebra | |||||||||
| · | multiplication | 3 · 4 means the multiplication of 3 by 4. | 7 · 8 = 56 | ||||||
| times | |||||||||
| arithmetic | |||||||||
| dot product | u · v means the dot product of vectors u and v | (1, 2, 5) · (3, 4, − 1) = 6 | |||||||
| dot | |||||||||
| vector algebra | |||||||||
| ÷ ⁄ | division | 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. | 2 ÷ 4 =.5 12 ⁄ 4 = 3 | ||||||
| divided by | |||||||||
| arithmetic | |||||||||
| ± | plus-minus | 6 ± 3 means both 6 + 3 and 6 - 3. | The equation x = 5 ± √ 4, has two solutions, x = 7 and x = 3. | ||||||
| plus or minus | |||||||||
| arithmetic | |||||||||
| plus-minus | 10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. | If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm. | |||||||
| plus or minus | |||||||||
| measurment | |||||||||
| ∓ | minus-plus | 6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5). | cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y). | ||||||
| minus or plus | |||||||||
| arithmetic | |||||||||
| √ | square root | √ x means the positive number whose square is x. | √ 4 = 2 | ||||||
| the principal square root of; square root | |||||||||
| real numbers | |||||||||
| complex square root | if z = r exp(i φ) is represented in polar coordinates with - π < φ ≤ π, then √ z = √ r exp(i φ /2). | √ (-1) = i | |||||||
| the complex square root of … square root | |||||||||
| complex numbers | |||||||||
| |…| | absolute value | | x | means the distance along the real line (or across the complex plane) between x and zero. | |3| = 3 |–5| = |5| | i | = 1 | 3 + 4 i | = 5 | ||||||
| absolute value 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 | |||||||||
| | | 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 | |||||||||
| ! | factorial | n! is the product 1 × 2×... × n. | 4! = 1 × 2 × 3 × 4 = 24 | ||||||
| factorial | |||||||||
| combinatorics | |||||||||
| T | transpose | Swap rows for columns | Aij = (AT) ji | ||||||
| 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 | |||||||||
| ⇒ → ⊃ | 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 ⇒ x 2 = 4 is true, but x 2 = 4 ⇒ x = 2 is in general false (since x could be − 2). | ||||||
| implies; if … then | |||||||||
| propositional logic | |||||||||
| ⇔ ↔ | 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) ⇔ Ax ≠ 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)). | 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 one 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 = ∅) | |||||||
| direct sum of | |||||||||
| Abstract algebra | |||||||||
| ∀ | universal quantification | ∀ x: P (x) means P (x) is true for all x. | ∀ n ∈ ℕ: n 2 ≥ 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 = 2 n. | ||||||
| there exists exactly one | |||||||||
| predicate logic | |||||||||
| : = ≡: ⇔ | definition | x: = y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence). P: ⇔ Q means P is defined to be logically equivalent to Q. | cosh x: = (1/2)(exp x + exp (− x)) A xor B: ⇔ (A ∨ B) ∧ (A ∧ B) | ||||||
| is defined as | |||||||||
| everywhere | |||||||||
| ≅ | congruence | △ ABC ≅ △ DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. | |||||||
| is congruent to | |||||||||
| geometry | |||||||||
| ≡ | congruence relation | a ≡ b (mod n) means a − b is divisible by n | 5 ≡ 11 (mod 3) | ||||||
| ... is congruent to... modulo... | |||||||||
| modular arithmetic | |||||||||
| {, } | set brackets | { a, b, c } means the set consisting of a, b, and c. | ℕ = { 1, 2, 3, …} | ||||||
| the set of … | |||||||||
| set theory | |||||||||
| {: } { | } | set builder notation | { x: P (x)} means the set of all x for which P (x) is true. { x | P (x)} is the same as { x: P (x)}. | { n ∈ ℕ: n 2 < 20} = { 1, 2, 3, 4} | ||||||
| the set of … such that | |||||||||
| set theory | |||||||||
| ∅ { } | empty set | ∅ means the set with no elements. { } means the same. | { n ∈ ℕ: 1 < n 2 < 4} = ∅ | ||||||
| the empty set | |||||||||
| set theory | |||||||||
| ∈ ∉ | set membership | a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. | (1/2)− 1 ∈ ℕ 2− 1 ∉ ℕ | ||||||
| is an element of; is not an element of | |||||||||
| everywhere, set theory | |||||||||
| ⊆ ⊂ | subset | (subset) A ⊆ B means every element of A is also element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.) | (A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ | ||||||
| is a subset of | |||||||||
| set theory | |||||||||
| ⊇ ⊃ | superset | A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.) | (A ∪ B) ⊇ B ℝ ⊃ ℚ | ||||||
| is a superset of | |||||||||
| set theory | |||||||||
| ∪ | set-theoretic union | (exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both. " A or B, but not both." (inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B. " A or B or both". | A ⊆ B ⇔ (A ∪ B) = B (inclusive) | ||||||
| the union of … and union | |||||||||
| set theory | |||||||||
| ∩ | set-theoretic intersection | A ∩ B means the set that contains all those elements that A and B have in common. | { x ∈ ℝ: x 2 = 1} ∩ ℕ = {1} | ||||||
| intersected with; intersect | |||||||||
| set theory | |||||||||
| Δ | symmetric difference | A Δ B means the set of elements in exactly one of A or B. | {1, 5, 6, 8} Δ {2, 5, 8} = {1, 2, 6} | ||||||
| symmetric difference | |||||||||
| set theory | |||||||||
| ∖ | set-theoretic complement | A ∖ B means the set that contains all those elements of A that are not in B. | {1, 2, 3, 4} ∖ {3, 4, 5, 6} = {1, 2} | ||||||
| minus; without | |||||||||
| set theory | |||||||||
| () | function application | f (x) means the value of the function f at the element x. | If f (x): = x 2, then f (3) = 32 = 9. | ||||||
| of | |||||||||
| set theory | |||||||||
| precedence grouping | Perform the operations inside the parentheses first. | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. | |||||||
| parentheses | |||||||||
| everywhere | |||||||||
| f: X → Y | function arrow | f: X → Y means the function f maps the set X into the set Y. | Let f: ℤ → ℕ be defined by f (x): = x 2. | ||||||
| from … to | |||||||||
| set theory | |||||||||
| o | function composition | f o g is the function, such that (f o g)(x) = f (g (x)). | if f (x): = 2 x, and g (x): = x + 3, then (f o g)(x) = 2(x + 3). | ||||||
| composed with | |||||||||
| set theory | |||||||||
| ℕ N | natural numbers | N means { 1, 2, 3,...}, but see the article on natural numbers for a different convention. | ℕ = {| a |: a ∈ ℤ, a ≠ 0} | ||||||
| N | |||||||||
| numbers | |||||||||
| ℤ Z | integers | ℤ means {..., − 3, − 2, − 1, 0, 1, 2, 3,...} and ℤ + means {1, 2, 3,...} = ℕ. | ℤ = { p, - p: p ∈ ℕ } ∪ {0} | ||||||
| Z | |||||||||
| numbers | |||||||||
| ℚ Q | rational numbers | ℚ means { p / q: p ∈ ℤ, q ∈ ℕ }. | 3.14000... ∈ ℚ π ∉ ℚ | ||||||
| Q | |||||||||
| numbers | |||||||||
| ℝ R | real numbers | ℝ means the set of real numbers. | π ∈ ℝ √ (− 1) ∉ ℝ | ||||||
| R | |||||||||
| numbers | |||||||||
| ℂ C | complex numbers | ℂ means { a + bi: a, b ∈ ℝ }. | i = √ (− 1) ∈ ℂ | ||||||
| C | |||||||||
| numbers | |||||||||
| arbitrary constant | C can be any number, most likely unknown; usually occurs when calculating antiderivatives. | if f(x) = 6 x ² + 4 x, then F(x) = 2 x ³ + 2 x ² + C, where F'(x) = f(x) | |||||||
| C | |||||||||
| integral calculus | |||||||||
| 𝕂 K | real or complex numbers | K means the statement holds substituting K for R and also for C. | 
because
and
 .
| ||||||
| K | |||||||||
| linear algebra | |||||||||
| ∞ | infinity | ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | limx→ 0 1/| x | = ∞ | ||||||
| infinity | |||||||||
| numbers | |||||||||
| ||…|| | norm | || x || is the norm of the element x of a normed vector space. | || x + y || ≤ || x || + || y || | ||||||
| norm of length of | |||||||||
| linear algebra | |||||||||
| ∑ | summation |  means a 1 + a 2 + … + an.
|  = 12 + 22 + 32 + 42
= 1 + 4 + 9 + 16 = 30
| ||||||
| sum over … from … to … of | |||||||||
| arithmetic | |||||||||
| ∏ | product |  means a 1 a 2··· an.
|  = (1+2)(2+2)(3+2)(4+2)
= 3 × 4 × 5 × 6 = 360
| ||||||
| product over … from … to … of | |||||||||
| arithmetic | |||||||||
| Cartesian product |  means the set of all (n+1)-tuples
(y 0, …, yn).
| 
| |||||||
| the Cartesian product of; the direct product of | |||||||||
| set theory | |||||||||
| ∐ | coproduct | ||||||||
| coproduct over … from … to … of | |||||||||
| category theory | |||||||||
| ′ • | derivative | f ′ (x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.
The dot notation indicates a time derivative. That is  .
| If f (x): = x 2, then f ′ (x) = 2 x | ||||||
| … prime derivative of | |||||||||
| calculus | |||||||||
| ∫ | indefinite integral or antiderivative | ∫ f (x) d x means a function whose derivative is f. | ∫ x 2 d x = x 3/3 + C | ||||||
| indefinite integral of the antiderivative of | |||||||||
| calculus | |||||||||
| definite integral | ∫ ab f (x) d x means the signed area between the x -axis and the graph of the function f between x = a and x = b. | ∫ 0 b x2 d x = b 3/3; | |||||||
| integral from … to … of … with respect to | |||||||||
| calculus | |||||||||
| ∇ | gradient | ∇ f (x1, …, x n) is the vector of partial derivatives (∂ f / ∂ x 1, …, ∂ f / ∂ xn). | If f (x, y, z): = 3 xy + z ², then ∇ f = (3 y, 3 x, 2 z) | ||||||
| del, nabla, gradient of | |||||||||
| calculus | |||||||||
| ∂ | partial differential | With f (x1, …, x n), ∂ f/∂ xi is the derivative of f with respect to xi, with all other variables kept constant. | If f (x, y): = x2y, then ∂ f /∂ x = 2xy | ||||||
| partial, d | |||||||||
| calculus | |||||||||
| boundary | ∂ M means the boundary of M | ∂ {x: ||x|| ≤ 2} = {x: ||x|| = 2} | |||||||
| boundary of | |||||||||
| topology | |||||||||
| ⊥ | perpendicular | x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. | If l ⊥ m and m ⊥ n then l || n. | ||||||
| is perpendicular to | |||||||||
| geometry | |||||||||
| bottom element | x = ⊥ means x is the smallest element. | ∀ x: x ∧ ⊥ = ⊥ | |||||||
| the bottom element | |||||||||
| lattice theory | |||||||||
| || | parallel | x || y means x is parallel to y. | If l || m and m ⊥ n then l ⊥ n. | ||||||
| is parallel to | |||||||||
| geometry | |||||||||
| ⊧ | entailment | A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true. | A ⊧ A ∨ A | ||||||
| entails | |||||||||
| model theory | |||||||||
| ⊢ | inference | x ⊢ y means y is derived from x. | A → B ⊢ B → A | ||||||
| infers or is derived from | |||||||||
| propositional logic, predicate logic | |||||||||
| ◅ | normal subgroup | N ◅ G means that N is a normal subgroup of group G. | Z (G) ◅ G | ||||||
| is a normal subgroup of | |||||||||
| group theory | |||||||||
| / | quotient group | G / H means the quotient of group G modulo its subgroup H. | {0, a, 2 a, b, b + a, b +2 a } / {0, b } = {{0, b }, { a, b + a }, {2 a, b +2 a }} | ||||||
| mod | |||||||||
| group theory | |||||||||
| quotient set | A /~ means the set of all ~ equivalence classes in A. | If we define ~ by x~y ⇔ x-y∈ Z, then R/~ = {{ x + n: n ∈ Z}: x ∈ (0, 1]} | |||||||
| mod | |||||||||
| set theory | |||||||||
| ≈ | isomorphism | G ≈ H means that group G is isomorphic to group H | Q / {1, − 1} ≈ V, where Q is the quaternion group and V is the Klein four-group. | ||||||
| is isomorphic to | |||||||||
| group theory | |||||||||
| approximately equal | x ≈ y means x is approximately equal to y | π ≈ 3.14159 | |||||||
| is approximately equal to | |||||||||
| everywhere | |||||||||
| ~ | same order of magnitude | m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈.) | 2 ~ 5 8 × 9 ~ 100 but π 2 ≈ 10 | ||||||
| roughly similar poorly approximates | |||||||||
| Approximation theory | |||||||||
| 〈, 〉 (|) ·: | inner product | 〈 x, y 〉 means the inner product of x and y as defined in an inner product space. For spatial vectors, the dot product notation, x · y is common. For matricies, the colon notation may be used. | The standard inner product between two vectors x = (2, 3) and y = (− 1, 5) is: 〈 x, y〉 = 2× − 1 + 3× 5 = 13
| ||||||
| inner product of | |||||||||
| vector algebra | |||||||||
| ⊗ | tensor product | V ⊗ U means the tensor product of V and U. | {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} | ||||||
| tensor product of | |||||||||
| linear algebra | |||||||||
| * | convolution | f * g means the convolution of f and g. | 
| ||||||
| convolution | |||||||||
| mean | is the mean (average value of xi).
|  .
| ||||||
| overbar | |||||||||
| statistics |
|
|