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Danh_sách_ký_hiệu_toán_học
| Symbol in HTML | Symbol in TeX | Name | Explanation | Examples |
|---|---|---|---|---|
| Read as | ||||
| Category | ||||
| ( ) {\displaystyle {\ \choose \ }} {\ \choose\ } | n choose k | ( n k ) = n ! / ( n − k ) ! k ! = ( n − k + 1 ) ⋯ ( n − 2 ) ⋅ ( n − 1 ) ⋅ n k ! {\displaystyle {\begin{pmatrix}n\\k\end{pmatrix}}={\frac {n!/(n-k)!}{k!}}={\frac {(n-k+1)\cdots (n-2)\cdot (n-1)\cdot n}{k!}}} means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements. (This may also be written as C(n, k), C(n; k), nCk, nCk, or ⟨ n k ⟩ {\displaystyle \left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle } .) | ( 36 5 ) = 36 ! / ( 36 − 5 ) ! 5 ! = 32 ⋅ 33 ⋅ 34 ⋅ 35 ⋅ 36 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 = 376992 {\displaystyle {\begin{pmatrix}36\\5\end{pmatrix}}={\frac {36!/(36-5)!}{5!}}={\frac {32\cdot 33\cdot 34\cdot 35\cdot 36}{1\cdot 2\cdot 3\cdot 4\cdot 5}}=376992} ( .5 7 ) = − 5.5 ⋅ − 4.5 ⋅ − 3.5 ⋅ − 2.5 ⋅ − 1.5 ⋅ − .5 ⋅ .5 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 = 33 2048 {\displaystyle {\begin{pmatrix}.5\\7\end{pmatrix}}={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={\frac {33}{2048}}\,\!} | |
| ( ( ) ) {\displaystyle \left(\!\!{\ \choose \ }\!\!\right)} \left(\!\!{\ \choose\ }\!\!\right) | u multichoose k | ( ( u k ) ) = ( u + k − 1 k ) = ( u + k − 1 ) ! / ( u − 1 ) ! k ! {\displaystyle \left(\!\!{u \choose k}\!\!\right)={u+k-1 \choose k}={\frac {(u+k-1)!/(u-1)!}{k!}}}
| ( ( − 5.5 7 ) ) = − 5.5 ⋅ − 4.5 ⋅ − 3.5 ⋅ − 2.5 ⋅ − 1.5 ⋅ − .5 ⋅ .5 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 = ( .5 7 ) = 33 2048 {\displaystyle \left(\!\!{-5.5 \choose 7}\!\!\right)={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={.5 \choose 7}={\frac {33}{2048}}\,\!} | |
| { … … {\displaystyle \left\{{\begin{array}{lr}\ldots \\\ldots \end{array}}\right.} \left\{ \begin{array}{lr} \ldots \\ \ldots \end{array}\right. | is defined as ... if ..., or as ... if ...; match ... with everywhere | f ( x ) = { a , if p ( x ) b , if q ( x ) {\displaystyle f(x)=\left\{{\begin{array}{rl}a,&{\text{if }}p(x)\\b,&{\text{if }}q(x)\end{array}}\right.} means the function f(x) is defined as a if the condition p(x) holds, or as b if the condition q(x) holds. (The body of a piecewise-defined function can have any finite number (not only just two) expression-condition pairs.) This symbol is also used in type theory for pattern matching the constructor of the value of an algebraic type. For example g ( n ) = match n with { x → a y → b {\displaystyle g(n)={\text{match }}n{\text{ with }}\left\{{\begin{array}{rl}x&\rightarrow a\\y&\rightarrow b\end{array}}\right.} does pattern matching on the function's arguments and means that g(x) is defined as a, and g(y) is defined as b. (A pattern matching can have any finite number (not only just two) pattern-expression pairs.) | | x | = { x , if x ≥ 0 − x , if x < 0 {\displaystyle |x|=\left\{{\begin{array}{rl}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0\end{array}}\right.} a + b = match b with { 0 → a S n → S ( a + n ) {\displaystyle a+b={\text{match }}b{\text{ with }}\left\{{\begin{array}{rl}0&\rightarrow a\\Sn&\rightarrow S(a+n)\end{array}}\right.} | |
| | … | {\displaystyle |\ldots |\!\,} | \ldots | \!\, | absolute value; modulus absolute value of; modulus of | |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 | |
Euclidean norm or Euclidean length or magnitude Euclidean norm of | |x| means the (Euclidean) length of vector x. | For x = (3,−4) | x | = 3 2 + ( − 4 ) 2 = 5 {\displaystyle |{\textbf {x}}|={\sqrt {3^{2}+(-4)^{2}}}=5} | ||
determinant of | |A| means the determinant of the matrix A | | 1 2 2 9 | = 5 {\displaystyle {\begin{vmatrix}1&2\\2&9\\\end{vmatrix}}=5} | ||
cardinality of; size of; order of | |X| means the cardinality of the set X. (# may be used instead as described below.) | |{3, 5, 7, 9}| = 4. | ||
| ‖ … ‖ {\displaystyle \|\ldots \|\!\,} \| \ldots \| \!\, | norm of; length of | ‖ x ‖ means the norm of the element x of a normed vector space.[10] | ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ | |
nearest integer to | ‖x‖ means the nearest integer to x. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).) | ‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3 | ||
| { , } {\displaystyle {\{\ ,\!\ \}}\!\,} {\{\ ,\!\ \}} \!\, | set brackets the set of ... | {a,b,c} means the set consisting of a, b, and c.[11] | ℕ = { 1, 2, 3, ... } | |
| { : } {\displaystyle \{\ :\ \}\!\,} \{\ :\ \} \!\, { | } {\displaystyle \{\ |\ \}\!\,} \{\ |\ \} \!\, { ; } {\displaystyle \{\ ;\ \}\!\,} \{\ ;\ \} \!\, | the set of ... such that | {x : P(x)} means the set of all x for which P(x) is true.[11] {x | P(x)} is the same as {x : P(x)}. | {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4 } | |
| ⌊ … ⌋ {\displaystyle \lfloor \ldots \rfloor \!\,} \lfloor \ldots \rfloor \!\, | floor; greatest integer; entier | ⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written [x], floor(x) or int(x).) | ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3 | |
| ⌈ … ⌉ {\displaystyle \lceil \ldots \rceil \!\,} \lceil \ldots \rceil \!\, | ceiling | ⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x. (This may also be written ceil(x) or ceiling(x).) | ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 | |
| ⌊ … ⌉ {\displaystyle \lfloor \ldots \rceil \!\,} \lfloor \ldots \rceil \!\, | nearest integer to | ⌊x⌉ means the nearest integer to x. (This may also be written [x], ||x||, nint(x) or Round(x).) | ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊−3.4⌉ = −3, ⌊4.49⌉ = 4, ⌊4.5⌉ = 5 | |
| [ : ] {\displaystyle [\ :\ ]\!\,} [\ :\ ] \!\, | the degree of | [K : F] means the degree of the extension K : F. | [ℚ(√2) : ℚ] = 2 [ℂ : ℝ] = 2 [ℝ : ℚ] = ∞ | |
| [ ] {\displaystyle [\ ]\!\,} [\ ] \!\, [ , ] {\displaystyle [\ ,\ ]\!\,} [\ ,\ ] \!\, [ , , ] {\displaystyle [\ ,\ ,\ ]\!\,} | the equivalence class of | [a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation. [a]R means the same, but with R as the equivalence relation. | Let a ~ b be true iff a ≡ b (mod 5). Then [2] = {..., −8, −3, 2, 7, ...}. | |
floor; greatest integer; entier | [x] means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written ⌊x⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) | [3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4 | ||
nearest integer to | [x] means the nearest integer to x. (This may also be written ⌊x⌉, ||x||, nint(x) or Round(x). Not to be confused with the floor function, as described above.) | [2] = 2, [2.6] = 3, [−3.4] = −3, [4.49] = 4 | ||
1 if true, 0 otherwise | [S] maps a true statement S to 1 and a false statement S to 0. | [0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0 | ||
image of ... under ... everywhere | f[X] means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(f). (This may also be written as f(X) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) | sin [ R ] = [ − 1 , 1 ] {\displaystyle \sin[\mathbb {R} ]=[-1,1]} | ||
closed interval | [ a , b ] = { x ∈ R : a ≤ x ≤ b } {\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}} . | 0 and 1/2 are in the interval [0,1]. | ||
the commutator of | [g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group). [a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra). | xy = x[x, y] (group theory). [AB, C] = A[B, C] + [A, C]B (ring theory). | ||
the triple scalar product of | [a, b, c] = a × b · c, the scalar product of a × b with c. | [a, b, c] = [b, c, a] = [c, a, b]. | ||
| ( ) {\displaystyle (\ )\!\,} (\ ) \!\, ( , ) {\displaystyle (\ ,\ )\!\,} (\ ,\ ) \!\, | function application of | f(x) means the value of the function f at the element x. | If f(x) := x2 − 5, then f(6) = 62 − 5 = 36 − 5=31. | |
image of ... under ... everywhere | f(X) means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(f). (This may also be written as f[X] if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) | sin ( R ) = [ − 1 , 1 ] {\displaystyle \sin(\mathbb {R} )=[-1,1]} | ||
precedence grouping parentheses everywhere | Perform the operations inside the parentheses first. | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. | ||
everywhere | An ordered list (or sequence, or horizontal vector, or row vector) of values. (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ⟨ ⟩ instead of parentheses.) | (a, b) is an ordered pair (or 2-tuple). (a, b, c) is an ordered triple (or 3-tuple). ( ) is the empty tuple (or 0-tuple). | ||
highest common factor; greatest common divisor; hcf; gcd number theory | (a, b) means the highest common factor of a and b. (This may also be written hcf(a, b) or gcd(a, b).) | (3, 7) = 1 (they are coprime); (15, 25) = 5. | ||
| ( , ) {\displaystyle (\ ,\ )\!\,} (\ ,\ ) \!\,(\ ,\ ) \!\, ] , [ {\displaystyle ]\ ,\ [\!\,} ]\ ,\ [ \!\,] | open interval | ( a , b ) = { x ∈ R : a < x < b } {\displaystyle (a,b)=\{x\in \mathbb {R} :a<x<b\}} . (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.) | 4 is not in the interval (4, 18). (0, +∞) equals the set of positive real numbers. | |
| ( , ] {\displaystyle (\ ,\ ]\!\,} (\ ,\ ] \!\, ] , ] {\displaystyle ]\ ,\ ]\!\,} \ ,\ ] \!\,] | half-open interval; left-open interval | ( a , b ] = { x ∈ R : a < x ≤ b } {\displaystyle (a,b]=\{x\in \mathbb {R} :a<x\leq b\}} . | (−1, 7] and (−∞, −1] | |
| [ , ) {\displaystyle [\ ,\ )\!\,} [\ ,\ ) \!\, [ , [ {\displaystyle [\ ,\ [\!\,} [\ ,\ [ \!\, | half-open interval; right-open interval | [ a , b ) = { x ∈ R : a ≤ x < b } {\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x<b\}} . | [4, 18) and [1, +∞) | |
| ⟨ ⟩ {\displaystyle \langle \ \rangle \!\,} \langle\ \rangle \!\, ⟨ , ⟩ {\displaystyle \langle \ ,\ \rangle \!\,} \langle\ ,\ \rangle \!\, | inner product of | ⟨u,v⟩ means the inner product of u and v, where u and v are members of an inner product space. Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span. There are many variants of the notation, such as ⟨u | v⟩ and (u | v), which are described below. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. | The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 | |
average average of | let S be a subset of N for example, ⟨ S ⟩ {\displaystyle \langle S\rangle } represents the average of all the elements in S. | for a time series :g(t) (t = 1, 2,...) we can define the structure functions Sq( τ {\displaystyle \tau } ): S q = ⟨ | g ( t + τ ) − g ( t ) | q ⟩ t {\displaystyle S_{q}=\langle |g(t+\tau )-g(t)|^{q}\rangle _{t}} | ||
the expectation value of | For a single discrete variable x {\displaystyle x} of a function f ( x ) {\displaystyle f(x)} , the expectation value of f ( x ) {\displaystyle f(x)} is defined as ⟨ f ( x ) ⟩ = ∑ x f ( x ) P ( x ) {\displaystyle \langle f(x)\rangle =\sum _{x}f(x)P(x)} , and for a single continuous variable the expectation value of f ( x ) {\displaystyle f(x)} is defined as ⟨ f ( x ) ⟩ = ∫ x f ( x ) P ( x ) {\displaystyle \langle f(x)\rangle =\int _{x}f(x)P(x)} ; where P ( x ) {\displaystyle P(x)} is the PDF of the variable x {\displaystyle x} .[12] | |||
(linear) span of; linear hull of | ⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. ⟨u1, u2, ...⟩ is shorthand for ⟨{u1, u2, ...}⟩.
| ⟨ ( 1 0 0 ) , ( 0 1 0 ) , ( 0 0 1 ) ⟩ = R 3 {\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}} . | ||
subgroup generated by a set the subgroup generated by | ⟨ S ⟩ {\displaystyle \langle S\rangle } means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S. ⟨ g 1 , g 2 , … ⟩ {\displaystyle \left\langle g_{1},g_{2},\dots \right\rangle } is shorthand for ⟨ { g 1 , g 2 , … } ⟩ {\displaystyle \left\langle \left\{g_{1},g_{2},\dots \right\}\right\rangle } . | In S3, ⟨ ( 1 2 ) ⟩ = { i d , ( 1 2 ) } {\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}} and ⟨ ( 1 2 3 ) ⟩ = { i d , ( 1 2 3 ) , ( 1 3 2 ) ) } {\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;3\;2))\}} . | ||
tuple; n-tuple; ordered pair/triple/etc; row vector; sequence everywhere | An ordered list (or sequence, or horizontal vector, or row vector) of values. (The notation (a,b) is often used as well.) | ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } is an ordered pair (or 2-tuple). ⟨ a , b , c ⟩ {\displaystyle \langle a,b,c\rangle } is an ordered triple (or 3-tuple). ⟨ ⟩ {\displaystyle \langle \rangle } is the empty tuple (or 0-tuple). | ||
| ⟨ | ⟩ {\displaystyle \langle \ |\ \rangle \!\,} \langle\ |\ \rangle \!\, ( | ) {\displaystyle (\ |\ )\!\,} (\ |\ ) \!\, | inner product of | ⟨u | v⟩ means the inner product of u and v, where u and v are members of an inner product space.[13] (u | v) means the same. Another variant of the notation is ⟨u, v⟩ which is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. |
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