A lộn ngược trong Toán học là gì

{\ \choose\ } [ 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 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!}}}

[when u is positive integer]
means reverse or rising binomial coefficient.

[ [ 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. 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 }}x0]=1, [2 {2,3,4}]=1, [5 {2,3,4}]=0

image

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

closed interval

order theory

[ 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].

commutator

the commutator of

group theory, ring theory

[g, h] = g1h1gh [or ghg1h1], 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].

triple scalar product

the triple scalar product of

vector calculus

[a, b, c] = a × b · c, the scalar product of a × b with c. [a, b, c] = [b, c, a] = [c, a, b].

Không thể phân tích cú pháp [lỗi cú pháp]: {\displaystyle [\] \!\,}
[\] \!\,

Không thể phân tích cú pháp [lỗi cú pháp]: {\displaystyle [\,\] \!\,}
[\,\] \!\,

function application

of

set theory

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

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.

tuple

tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence

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

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.

Không thể phân tích cú pháp [lỗi cú pháp]: {\displaystyle [\,\] \!\,}
[\,\] \!\,[\,\] \!\,

] [ {\displaystyle ]\,\ [\!\,}
]\,\ [ \!\,]

open interval

open interval

order theory

[ a , b ] = { x R : a < x < b } {\displaystyle [a,b]=\{x\in \mathbb {R} :a