In mathematics , the power set (or powerset ) of a set S is the set of all subsets of S , including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set . The powerset of S is variously denoted as P ( S ) , đ« ( S ) , P ( S ) , P ( S ) {\displaystyle \mathbb {P} (S)} , or 2 . Any subset of P ( S ) is called a family of sets over S .
113-422: If S is the set { x , y , z } , then all the subsets of S are and hence the power set of S is {{}, { x }, { y }, { z }, { x , y }, { x , z }, { y , z }, { x , y , z }} . If S is a finite set with the cardinality | S | = n (i.e., the number of all elements in the set S is n ), then the number of all the subsets of S is | P ( S ) | = 2 . This fact as well as
226-414: A → b {\displaystyle h:a\rightarrow b} to precomposition by h , so a function h ∗ : C ( b , c ) → C ( a , c ) {\displaystyle h^{*}:C(b,c)\rightarrow C(a,c)} , which takes morphisms from b to c and takes them to morphisms from a to c , through b via h . In category theory and
339-532: A + b ) ( n ) = ∑ k = 0 n ( n k ) a ( n − k ) b ( k ) . {\displaystyle (a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.} The case c = 0 recovers the usual binomial theorem. More generally, a sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }} of polynomials
452-472: A and b , the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a , a square of side b , and two rectangles with sides a and b . With n = 3 , the theorem states that a cube of side a + b can be cut into a cube of side a , a cube of side b , three a Ă a Ă b rectangular boxes, and three a Ă b Ă b rectangular boxes. In calculus , this picture also gives
565-666: A binomial . According to the theorem, it is possible to expand the polynomial ( x + y ) into a sum involving terms of the form ax y , where the exponents b and c are nonnegative integers with b + c = n , and the coefficient a of each term is a specific positive integer depending on n and b . For example, for n = 4 , ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 . {\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.} The coefficient
678-458: A function from A {\displaystyle A} to B {\displaystyle B} that is both injective and surjective . Such sets are said to be equipotent , equipollent , or equinumerous . For example, the set E = { 0 , 2 , 4 , 6 , ... } {\displaystyle E=\{0,2,4,6,{\text{...}}\}} of non-negative even numbers has
791-547: A in the term of ax y is known as the binomial coefficient ( n b ) {\displaystyle {\tbinom {n}{b}}} or ( n c ) {\displaystyle {\tbinom {n}{c}}} (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle . These numbers also occur in combinatorics , where ( n b ) {\displaystyle {\tbinom {n}{b}}} gives
904-416: A , then this picture shows the infinitesimal change in the volume of an n -dimensional hypercube , ( x + Δ x ) n , {\displaystyle (x+\Delta x)^{n},} where the coefficient of the linear term (in Δ x {\displaystyle \Delta x} ) is n x n − 1 , {\displaystyle nx^{n-1},}
1017-427: A cardinality of any proper class P {\displaystyle P} , in particular This definition is natural since it agrees with the axiom of limitation of size which implies bijection between V {\displaystyle V} and any proper class. Binomial theorem In elementary algebra , the binomial theorem (or binomial expansion ) describes the algebraic expansion of powers of
1130-398: A geometric proof of the derivative ( x n ) ′ = n x n − 1 : {\displaystyle (x^{n})'=nx^{n-1}:} if one sets a = x {\displaystyle a=x} and b = Δ x , {\displaystyle b=\Delta x,} interpreting b as an infinitesimal change in
1243-401: A one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of Cantor's diagonal argument , he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. While
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#17327724829501356-545: A one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite . Cantor introduced the cardinal numbers, and showedâaccording to his bijection-based definition of sizeâthat some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers ( ℵ 0 {\displaystyle \aleph _{0}} ). One of Cantor's most important results
1469-417: A set may also be called its size , when no confusion with other notions of size is possible. When two sets, A {\displaystyle A} and B {\displaystyle B} , have the same cardinality, it is usually written as | A | = | B | {\displaystyle |A|=|B|} ; however, if referring to
1582-434: A set": Assuming the axiom of choice , the cardinalities of the infinite sets are denoted For each ordinal α {\displaystyle \alpha } , ℵ α + 1 {\displaystyle \aleph _{\alpha +1}} is the least cardinal number greater than ℵ α {\displaystyle \aleph _{\alpha }} . The cardinality of
1695-437: A similar argument, N {\displaystyle \mathbb {N} } has cardinality strictly less than the cardinality of the set R {\displaystyle \mathbb {R} } of all real numbers . For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof . In the above section, "cardinality" of a set was defined functionally. In other words, it
1808-528: A specific negative value of y : ( x − 2 ) 3 = x 3 − 3 x 2 ( 2 ) + 3 x ( 2 ) 2 − 2 3 = x 3 − 6 x 2 + 12 x − 8. {\displaystyle {\begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\\&=x^{3}-6x^{2}+12x-8.\end{aligned}}} For positive values of
1921-487: A specific positive value of y : ( x + 2 ) 3 = x 3 + 3 x 2 ( 2 ) + 3 x ( 2 ) 2 + 2 3 = x 3 + 6 x 2 + 12 x + 8. {\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}} A simple example with
2034-501: A work by Al-Karaji , quoted by Al-Samaw'al in his "al-Bahir". Al-Karaji described the triangular pattern of the binomial coefficients and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction . The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in
2147-456: Is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras, this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem ). The power set of a set S forms an abelian group when it is considered with the operation of symmetric difference (with
2260-445: Is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum ). The power set of a set S , together with the operations of union , intersection and complement , is a ÎŁ-algebra over S and can be viewed as the prototypical example of a Boolean algebra . In fact, one can show that any finite Boolean algebra
2373-560: Is a finite set , then a recursive definition of P ( S ) proceeds as follows: In words: The set of subsets of S of cardinality less than or equal to Îș is sometimes denoted by P Îș ( S ) or [ S ] , and the set of subsets with cardinality strictly less than Îș is sometimes denoted P < Îș ( S ) or [ S ] . Similarly, the set of non-empty subsets of S might be denoted by P â„1 ( S ) or P ( S ) . A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective,
SECTION 20
#17327724829502486-501: Is a bijection. This is no longer true for infinite A {\displaystyle A} and B {\displaystyle B} . For example, the function g {\displaystyle g} from N {\displaystyle \mathbb {N} } to E {\displaystyle E} , defined by g ( n ) = 4 n {\displaystyle g(n)=4n}
2599-807: Is a nonnegative integer, the binomial coefficients for k > r are zero, so this equation reduces to the usual binomial theorem, and there are at most r + 1 nonzero terms. For other values of r , the series typically has infinitely many nonzero terms. For example, r = 1/2 gives the following series for the square root: 1 + x = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 − 5 128 x 4 + 7 256 x 5 − ⋯ . {\displaystyle {\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots .} Taking r = −1 ,
2712-610: Is a positive integer known as a binomial coefficient , defined as ( n k ) = n ! k ! ( n − k ) ! = n ( n − 1 ) ( n − 2 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ( k − 2 ) ⋯ 2 ⋅ 1 . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1}}.} This formula
2825-508: Is also a polynomial in x and y , and [ ( x + y ) n + 1 ] j , k = [ ( x + y ) n ] j − 1 , k + [ ( x + y ) n ] j , k − 1 , {\displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},} since if j + k = n + 1 , then ( j â 1) + k = n and j + ( k â 1) = n . Now,
2938-635: Is also referred to as the binomial formula or the binomial identity . Using summation notation , it can be written more concisely as ( x + y ) n = ∑ k = 0 n ( n k ) x n − k y k = ∑ k = 0 n ( n k ) x k y n − k . {\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.} The final expression follows from
3051-408: Is always an algebraic lattice , and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets. However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize
3164-499: Is an injective function from N {\displaystyle \mathbb {N} } to P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} , and it can be shown that no function from N {\displaystyle \mathbb {N} } to P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} can be bijective (see picture). By
3277-459: Is another name for a k âelements subset, so the number of combinations , denoted as C( n , k ) (also called binomial coefficient ) is a number of subsets with k elements in a set with n elements; in other words it's the number of sets with k elements which are elements of the power set of a set with n elements. For example, the power set of a set with three elements, has: Using this relationship, we can compute | 2 | using
3390-470: Is apparent by considering, for instance, the tangent function , which provides a one-to-one correspondence between the interval (âÂœÏ, ÂœÏ) and R (see also Hilbert's paradox of the Grand Hotel ). The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano introduced the space-filling curves , curved lines that twist and turn enough to fill
3503-543: Is binomial if and only if its basis operator is a Delta operator . Writing E a {\displaystyle E^{a}} for the shift by a {\displaystyle a} operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference I − E − c {\displaystyle I-E^{-c}} for c > 0 {\displaystyle c>0} ,
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3616-461: Is called a presheaf . Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed ) and has an object Ω , called a subobject classifier . Although the term "power object" is sometimes used synonymously with exponential object Y , in topos theory Y
3729-429: Is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S , although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it. The first of these results
3842-499: Is equivalent to the statement that | A | ≤ | B | {\displaystyle |A|\leq |B|} or | B | ≤ | A | {\displaystyle |B|\leq |A|} for every A {\displaystyle A} and B {\displaystyle B} . A {\displaystyle A} has cardinality strictly less than
3955-1044: Is given by the formula ( n k ) = n ! k ! ( n − k ) ! , {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n ! . Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 = ∏ ℓ = 1 k n − ℓ + 1 ℓ = ∏ ℓ = 0 k − 1 n − ℓ k − ℓ {\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}} with k factors in both
4068-475: Is injective, but not surjective since 2, for instance, is not mapped to, and h {\displaystyle h} from N {\displaystyle \mathbb {N} } to E {\displaystyle E} , defined by h ( n ) = n − ( n mod 2 ) {\displaystyle h(n)=n-(n{\text{ mod }}2)} (see: modulo operation )
4181-592: Is invertible, and ‖ y / x ‖ < 1 . A version of the binomial theorem is valid for the following Pochhammer symbol -like family of polynomials: for a given real constant c , define x ( 0 ) = 1 {\displaystyle x^{(0)}=1} and x ( n ) = ∏ k = 1 n [ x + ( k − 1 ) c ] {\displaystyle x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]} for n > 0. {\displaystyle n>0.} Then (
4294-419: Is located at the first from the right of this sequence and y is at the second from the right, and 1 in the sequence means the element of S corresponding to the position of it in the sequence exists in the subset of S for the sequence while 0 means it does not. For the whole power set of S , we get: Such an injective mapping from P ( S ) to integers is arbitrary, so this representation of all
4407-423: Is more common; the case of having both is relatively rare. One class that does have both is that of multigraphs . Given two multigraphs G and H , a homomorphism h : G â H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set H of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively
4520-434: Is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory , if ZFC is consistent. Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this
4633-407: Is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFCâprovided that ZFC is consistent. For more detail, see § Cardinality of the continuum below. If the axiom of choice holds,
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4746-978: Is obtained by substituting 1 for y , so that it involves only a single variable . In this form, the formula reads ( x + 1 ) n = ( n 0 ) x 0 + ( n 1 ) x 1 + ( n 2 ) x 2 + ⋯ + ( n n − 1 ) x n − 1 + ( n n ) x n = ∑ k = 0 n ( n k ) x k . ) {\displaystyle {\begin{aligned}(x+1)^{n}&={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n}\\[4mu]&=\sum _{k=0}^{n}{n \choose k}x^{k}.{\vphantom {\Bigg )}}\end{aligned}}} Here are
4859-409: Is related to binomials for the following reason: if we write ( x + y ) as a product ( x + y ) ( x + y ) ( x + y ) ⋯ ( x + y ) , {\displaystyle (x+y)(x+y)(x+y)\cdots (x+y),} then, according to the distributive law , there will be one term in the expansion for each choice of either x or y from each of
4972-808: Is required to be Ω . There is both a covariant and contravariant power set functor , P : Set â Set and P : Set â Set . The covariant functor is defined more simply. as the functor which sends a set S to P ( S ) and a morphism f : S â T (here, a function between sets) to the image morphism. That is, for A = { x 1 , x 2 , . . . } ∈ P ( S ) , P f ( A ) = { f ( x 1 ) , f ( x 2 ) , . . . } ∈ P ( T ) {\displaystyle A=\{x_{1},x_{2},...\}\in {\mathsf {P}}(S),{\mathsf {P}}f(A)=\{f(x_{1}),f(x_{2}),...\}\in {\mathsf {P}}(T)} . Elsewhere in this article,
5085-739: Is said to be of binomial type if An operator Q {\displaystyle Q} on the space of polynomials is said to be the basis operator of the sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }} if Q p 0 = 0 {\displaystyle Qp_{0}=0} and Q p n = n p n − 1 {\displaystyle Qp_{n}=np_{n-1}} for all n ⩾ 1 {\displaystyle n\geqslant 1} . A sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }}
5198-546: Is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither g {\displaystyle g} nor h {\displaystyle h} can challenge | E | = | N | {\displaystyle |E|=|\mathbb {N} |} , which was established by the existence of f {\displaystyle f} . A {\displaystyle A} has cardinality less than or equal to
5311-1176: Is the Pochhammer symbol , here standing for a falling factorial . This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with | x | > | y | , and r is any complex number, one has ( x + y ) r = ∑ k = 0 ∞ ( r k ) x r − k y k = x r + r x r − 1 y + r ( r − 1 ) 2 ! x r − 2 y 2 + r ( r − 1 ) ( r − 2 ) 3 ! x r − 3 y 3 + ⋯ . {\displaystyle {\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}} When r
5424-488: Is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step. Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series . In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using
5537-816: The cardinal number of an individual set A {\displaystyle A} , it is simply denoted | A | {\displaystyle |A|} , with a vertical bar on each side; this is the same notation as absolute value , and the meaning depends on context. The cardinal number of a set A {\displaystyle A} may alternatively be denoted by n ( A ) {\displaystyle n(A)} , A {\displaystyle A} , card ( A ) {\displaystyle \operatorname {card} (A)} , or # A {\displaystyle \#A} . A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or
5650-512: The complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine . According to De Moivre's formula, cos ( n x ) + i sin ( n x ) = ( cos x + i sin x ) n . {\displaystyle \cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.} Using
5763-496: The definition of the derivative via a difference quotient and taking limits means that the higher order terms, ( Δ x ) 2 {\displaystyle (\Delta x)^{2}} and higher, become negligible, and yields the formula ( x n ) ′ = n x n − 1 , {\displaystyle (x^{n})'=nx^{n-1},} interpreted as If one integrates this picture, which corresponds to applying
SECTION 50
#17327724829505876-650: The fundamental theorem of calculus , one obtains Cavalieri's quadrature formula , the integral ∫ x n − 1 d x = 1 n x n {\displaystyle \textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}} â see proof of Cavalieri's quadrature formula for details. The coefficients that appear in the binomial expansion are called binomial coefficients . These are usually written ( n k ) , {\displaystyle {\tbinom {n}{k}},} and pronounced " n choose k ". The coefficient of x y
5989-411: The isomorphism with the binary representations of numbers from 0 to 2 â 1 , with n being the number of elements in the set S or | S | = n . First, the enumerated set { ( x , 1), ( y , 2), ( z , 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as { x , y } = 011 (2) ; x of S
6102-474: The law of trichotomy holds for cardinality. Thus we can make the following definitions: Our intuition gained from finite sets breaks down when dealing with infinite sets . In the late 19th century Georg Cantor , Gottlob Frege , Richard Dedekind and others rejected the view that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel . Indeed, Dedekind defined an infinite set as one that can be placed into
6215-474: The n th derivative of a product of two functions in a form similar to that of the binomial theorem: ( f g ) ( n ) ( x ) = ∑ k = 0 n ( n k ) f ( n − k ) ( x ) g ( k ) ( x ) . {\displaystyle (fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).} Here,
6328-683: The natural numbers is denoted aleph-null ( ℵ 0 {\displaystyle \aleph _{0}} ), while the cardinality of the real numbers is denoted by " c {\displaystyle {\mathfrak {c}}} " (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum . Cantor showed, using the diagonal argument , that c > ℵ 0 {\displaystyle {\mathfrak {c}}>\aleph _{0}} . We can show that c = 2 ℵ 0 {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}} , this also being
6441-404: The pre image morphism, so that if f ( A ) = B ⊆ T , P ¯ f ( B ) = A {\displaystyle f(A)=B\subseteq T,{\overline {\mathsf {P}}}f(B)=A} . This is because a general functor C ( − , c ) {\displaystyle {\text{C}}(-,c)} takes a morphism h :
6554-573: The 13th century mathematical works of Yang Hui and also Chu Shih-Chieh . Yang Hui attributes the method to a much earlier 11th century text of Jia Xian , although those writings are now also lost. In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express ( 1 + x ) n {\displaystyle (1+x)^{n}} in terms of ( 1 + x ) n − 1 {\displaystyle (1+x)^{n-1}} , via "Pascal's triangle". Blaise Pascal studied
6667-465: The Indian mathematicians probably knew how to express this as a quotient n ! ( n − k ) ! k ! {\textstyle {\frac {n!}{(n-k)!k!}}} , and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara . The first known formulation of the binomial theorem and the table of binomial coefficients appears in
6780-491: The area of the n faces, each of dimension n − 1 : ( x + Δ x ) n = x n + n x n − 1 Δ x + ( n 2 ) x n − 2 ( Δ x ) 2 + ⋯ . {\displaystyle (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .} Substituting this into
6893-1475: The binomial theorem this is equal to ( x 1 + y 1 ) n 1 ⋯ ( x d + y d ) n d = ∑ k 1 = 0 n 1 ⋯ ∑ k d = 0 n d ( n 1 k 1 ) x 1 k 1 y 1 n 1 − k 1 … ( n d k d ) x d k d y d n d − k d . {\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.} This may be written more concisely, by multi-index notation , as ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.} The general Leibniz rule gives
SECTION 60
#17327724829507006-805: The binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos( nx ) and sin( nx ) . For example, since ( cos x + i sin x ) 2 = cos 2 x + 2 i cos x sin x − sin 2 x = ( cos 2 x − sin 2 x ) + i ( 2 cos x sin x ) , {\displaystyle \left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x=(\cos ^{2}x-\sin ^{2}x)+i(2\cos x\sin x),} But De Moivre's formula identifies
7119-458: The binomials of the product. For example, there will only be one term x , corresponding to choosing x from each binomial. However, there will be several terms of the form x y , one for each way of choosing exactly two binomials to contribute a y . Therefore, after combining like terms , the coefficient of x y will be equal to the number of ways to choose exactly 2 elements from an n -element set. Expanding ( x + y ) yields
7232-774: The cardinalities of unions and intersections are related by the following equation: Here V {\displaystyle V} denote a class of all sets, and Ord {\displaystyle {\mbox{Ord}}} denotes the class of all ordinal numbers. We use the intersection of a class which is defined by ( x ∈ ⋂ Q ) ⟺ ( ∀ q ∈ Q : x ∈ q ) {\displaystyle (x\in \bigcap Q)\iff (\forall q\in Q:x\in q)} , therefore ⋂ ∅ = V {\displaystyle \bigcap \emptyset =V} . In this case This definition allows also obtain
7345-656: The cardinality of B {\displaystyle B} , if there exists an injective function from A {\displaystyle A} into B {\displaystyle B} . If | A | ≤ | B | {\displaystyle |A|\leq |B|} and | B | ≤ | A | {\displaystyle |B|\leq |A|} , then | A | = | B | {\displaystyle |A|=|B|} (a fact known as SchröderâBernstein theorem ). The axiom of choice
7458-659: The cardinality of B {\displaystyle B} , if there is an injective function, but no bijective function, from A {\displaystyle A} to B {\displaystyle B} . For example, the set N {\displaystyle \mathbb {N} } of all natural numbers has cardinality strictly less than its power set P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} , because g ( n ) = { n } {\displaystyle g(n)=\{n\}}
7571-474: The cardinality of a finite set is simply comparable to its number of elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite). Two sets have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A {\displaystyle A} to B {\displaystyle B} , that is,
7684-474: The cardinality of the set of all subsets of the natural numbers. The continuum hypothesis says that ℵ 1 = 2 ℵ 0 {\displaystyle \aleph _{1}=2^{\aleph _{0}}} , i.e. 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} is the smallest cardinal number bigger than ℵ 0 {\displaystyle \aleph _{0}} , i.e. there
7797-415: The case where x and y are complex numbers. For this version, one should again assume | x | > | y | and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius | x | centered at x . The generalized binomial theorem is valid also for elements x and y of a Banach algebra as long as xy = yx , and x
7910-407: The division of things into parts repeated without limit. In Euclid's Elements , commensurability was described as the ability to compare the length of two line segments, a and b , as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both a and b . But with the discovery of irrational numbers , it was seen that even
8023-415: The empty set as the identity element and each set being its own inverse), and a commutative monoid when considered with the operation of intersection (with the entire set S as the identity element). It can hence be shown, by proving the distributive laws , that the power set considered together with both of these operations forms a Boolean ring . In set theory , X is the notation representing
8136-510: The eponymous triangle comprehensively in his TraitĂ© du triangle arithmĂ©tique . However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, NiccolĂČ Fontana Tartaglia , and Simon Stevin . By the early 17th century, some specific cases of the generalized binomial theorem, such as for n = 1 2 {\displaystyle n={\frac {1}{2}}} , can be found in
8249-705: The equality holds for a given n ; we will prove it for n + 1 . For j , k â„ 0 , let [ f ( x , y )] j , k denote the coefficient of x y in the polynomial f ( x , y ) . By the inductive hypothesis, ( x + y ) is a polynomial in x and y such that [( x + y ) ] j , k is ( n k ) {\displaystyle {\tbinom {n}{k}}} if j + k = n , and 0 otherwise. The identity ( x + y ) n + 1 = x ( x + y ) n + y ( x + y ) n {\displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}} shows that ( x + y )
8362-951: The expansion of ( x + y ) on the right side in the n th row (numbered so that the top row is the 0th row): An example illustrating the last two points: ( x + y ) 3 = x x x + x x y + x y x + x y y + y x x + y x y + y y x + y y y ( 2 3 terms ) = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( 3 + 1 terms ) {\displaystyle {\begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{\text{ terms}})\\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{\text{ terms}})\end{aligned}}} with 1 + 3 + 3 + 1 = 2 3 {\displaystyle 1+3+3+1=2^{3}} . A simple example with
8475-2715: The first few cases of the binomial theorem: ( x + y ) 0 = 1 , ( x + y ) 1 = x + y , ( x + y ) 2 = x 2 + 2 x y + y 2 , ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 , ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 , ( x + y ) 5 = x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 + 5 x y 4 + y 5 , ( x + y ) 6 = x 6 + 6 x 5 y + 15 x 4 y 2 + 20 x 3 y 3 + 15 x 2 y 4 + 6 x y 5 + y 6 , ( x + y ) 7 = x 7 + 7 x 6 y + 21 x 5 y 2 + 35 x 4 y 3 + 35 x 3 y 4 + 21 x 2 y 5 + 7 x y 6 + y 7 , ( x + y ) 8 = x 8 + 8 x 7 y + 28 x 6 y 2 + 56 x 5 y 3 + 70 x 4 y 4 + 56 x 3 y 5 + 28 x 2 y 6 + 8 x y 7 + y 8 . {\displaystyle {\begin{aligned}(x+y)^{0}&=1,\\[8pt](x+y)^{1}&=x+y,\\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.\end{aligned}}} In general, for
8588-407: The formula ( n k 1 , k 2 , … , k m ) = n ! k 1 ! ⋅ k 2 ! ⋯ k m ! . {\displaystyle {\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.} Combinatorially,
8701-602: The formula: | 2 S | = ∑ k = 0 | S | ( | S | k ) {\displaystyle \left|2^{S}\right|=\sum _{k=0}^{|S|}{\binom {|S|}{k}}} Therefore, one can deduce the following identity, assuming | S | = n : | 2 S | = 2 n = ∑ k = 0 n ( n k ) {\displaystyle \left|2^{S}\right|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}} If S
8814-2779: The generalized binomial series gives the geometric series formula , valid for | x | < 1 : ( 1 + x ) − 1 = 1 1 + x = 1 − x + x 2 − x 3 + x 4 − x 5 + ⋯ . {\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .} More generally, with r = â s , we have for | x | < 1 : 1 ( 1 + x ) s = ∑ k = 0 ∞ ( − s k ) x k = ∑ k = 0 ∞ ( s + k − 1 k ) ( − 1 ) k x k . {\displaystyle {\frac {1}{(1+x)^{s}}}=\sum _{k=0}^{\infty }{-s \choose k}x^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}x^{k}.} So, for instance, when s = 1/2 , 1 1 + x = 1 − 1 2 x + 3 8 x 2 − 5 16 x 3 + 35 128 x 4 − 63 256 x 5 + ⋯ . {\displaystyle {\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots .} Replacing x with -x yields: 1 ( 1 − x ) s = ∑ k = 0 ∞ ( s + k − 1 k ) ( − 1 ) k ( − x ) k = ∑ k = 0 ∞ ( s + k − 1 k ) x k . {\displaystyle {\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}(-x)^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.} So, for instance, when s = 1/2 , we have for | x | < 1 : 1 1 − x = 1 + 1 2 x + 3 8 x 2 + 5 16 x 3 + 35 128 x 4 + 63 256 x 5 + ⋯ . {\displaystyle {\frac {1}{\sqrt {1-x}}}=1+{\frac {1}{2}}x+{\frac {3}{8}}x^{2}+{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}+{\frac {63}{256}}x^{5}+\cdots .} The generalized binomial theorem can be extended to
8927-453: The idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra. The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion,
9040-402: The infinite set of all rational numbers was not enough to describe the length of every possible line segment. Still, there was no concept of infinite sets as something that had cardinality. To better understand infinite sets, a notion of cardinality was formulated c. 1880 by Georg Cantor , the originator of set theory . He examined the process of equating two sets with bijection ,
9153-659: The left side with ( cos x + i sin x ) 2 = cos ( 2 x ) + i sin ( 2 x ) {\displaystyle (\cos x+i\sin x)^{2}=\cos(2x)+i\sin(2x)} , so cos ( 2 x ) = cos 2 x − sin 2 x and sin ( 2 x ) = 2 cos x sin x , {\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,} which are
9266-447: The manipulation of numbers without reference to a specific group of things or events. From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing. The ancient Greek notion of infinity also considered
9379-441: The multinomial coefficient ( n k 1 , ⋯ , k m ) {\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}} counts the number of different ways to partition an n -element set into disjoint subsets of sizes k 1 , ..., k m . When working in more dimensions, it is often useful to deal with products of binomial expressions. By
9492-402: The number of different combinations (i.e. subsets) of b elements that can be chosen from an n -element set . Therefore ( n b ) {\displaystyle {\tbinom {n}{b}}} is usually pronounced as " n choose b ". Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned
9605-457: The number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaáž„ĆÄstra by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution. The commentator Halayudha from the 10th century AD explains this method. By the 6th century AD,
9718-463: The numerator and denominator of the fraction . Although this formula involves a fraction, the binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} is actually an integer . The binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} can be interpreted as the number of ways to choose k elements from an n -element set. This
9831-1081: The ordinary derivative for c = 0 {\displaystyle c=0} , and the forward difference E − c − I {\displaystyle E^{-c}-I} for c < 0 {\displaystyle c<0} . The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 + k 2 + ⋯ + k m = n ( n k 1 , k 2 , … , k m ) x 1 k 1 x 2 k 2 ⋯ x m k m , {\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},} where
9944-434: The position of binary digit sequences.) The enumeration is possible even if S has an infinite cardinality (i.e., the number of elements in S is infinite), such as the set of integers or rationals, but not possible for example if S is the set of real numbers, in which case we cannot enumerate all irrational numbers. The binomial theorem is closely related to the power set. A k âelements combination from some set
10057-411: The power set was defined as the set of functions of S into the set with 2 elements. Formally, this defines a natural isomorphism P ¯ ≅ Set ( − , 2 ) {\displaystyle {\overline {\mathsf {P}}}\cong {\text{Set}}(-,2)} . The contravariant power set functor is different from the covariant version in that it sends f to
10170-401: The previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, ( n k ) = ( n n − k ) . {\textstyle {\binom {n}{k}}={\binom {n}{n-k}}.} A simple variant of the binomial formula
10283-405: The reason of the notation 2 denoting the power set P ( S ) are demonstrated in the below. Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set
10396-777: The right hand side is ( n k ) + ( n k − 1 ) = ( n + 1 k ) , {\displaystyle {\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{k}},} by Pascal's identity . On the other hand, if j + k â n + 1 , then ( j â 1) + k â n and j + ( k â 1) â n , so we get 0 + 0 = 0 . Thus ( x + y ) n + 1 = ∑ k = 0 n + 1 ( n + 1 k ) x n + 1 − k y k , {\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},} which
10509-966: The same cardinality as the set N = { 0 , 1 , 2 , 3 , ... } {\displaystyle \mathbb {N} =\{0,1,2,3,{\text{...}}\}} of natural numbers , since the function f ( n ) = 2 n {\displaystyle f(n)=2n} is a bijection from N {\displaystyle \mathbb {N} } to E {\displaystyle E} (see picture). For finite sets A {\displaystyle A} and B {\displaystyle B} , if some bijection exists from A {\displaystyle A} to B {\displaystyle B} , then each injective or surjective function from A {\displaystyle A} to B {\displaystyle B}
10622-500: The same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen as early as 40 000 years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian mathematics and
10735-404: The set of all functions from Y to X . As " 2 " can be defined as {0, 1} (see, for example, von Neumann ordinals ), 2 (i.e., {0, 1} ) is the set of all functions from S to {0, 1} . As shown above , 2 and the power set of S , P ( S ) , are considered identical set-theoretically. This equivalence can be applied to the example above , in which S = { x , y , z } , to get
10848-645: The sets A = { 1 , 2 , 3 } {\displaystyle A=\{1,2,3\}} and B = { 2 , 4 , 6 } {\displaystyle B=\{2,4,6\}} are the same size as they each contain 3 elements . Beginning in the late 19th century, this concept was generalized to infinite sets , which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two notions often used when referring to cardinality: one which compares sets directly using bijections and injections , and another which uses cardinal numbers . The cardinality of
10961-516: The special case of the binomial theorem for exponent n = 2 {\displaystyle n=2} . Greek mathematician Diophantus cubed various binomials, including x − 1 {\displaystyle x-1} . Indian mathematician Aryabhata 's method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent n = 3 {\displaystyle n=3} . Binomial coefficients, as combinatorial quantities expressing
11074-424: The subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0, 1} = 2 , there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way. Certain classes of algebras enjoy both of these properties. The first property
11187-427: The subgraphs of G as the multigraph Ω , called the power object of G . What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s , t : E â V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary
11300-480: The subsets of S is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., { ( y , 1), ( z , 2), ( x , 3) } can be used to construct another injective mapping from P ( S ) to the integers without changing the number of one-to-one correspondences.) However, such finite binary representation is only possible if S can be enumerated. (In this example, x , y , and z are enumerated with 1 , 2 , and 3 respectively as
11413-1468: The sum of the 2 products of the form e 1 e 2 ... e n where each e i is x or y . Rearranging factors shows that each product equals x y for some k between 0 and n . For a given k , the following are proved equal in succession: This proves the binomial theorem. The coefficient of xy in ( x + y ) 3 = ( x + y ) ( x + y ) ( x + y ) = x x x + x x y + x y x + x y y _ + y x x + y x y _ + y y x _ + y y y = x 3 + 3 x 2 y + 3 x y 2 _ + y 3 {\displaystyle {\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}} equals ( 3 2 ) = 3 {\displaystyle {\tbinom {3}{2}}=3} because there are three x , y strings of length 3 with exactly two y 's, namely, x y y , y x y , y y x , {\displaystyle xyy,\;yxy,\;yyx,} corresponding to
11526-483: The summation is taken over all sequences of nonnegative integer indices k 1 through k m such that the sum of all k i is n . (For each term in the expansion, the exponents must add up to n ). The coefficients ( n k 1 , ⋯ , k m ) {\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}} are known as multinomial coefficients, and can be computed by
11639-426: The superscript ( n ) indicates the n th derivative of a function, f ( n ) ( x ) = d n d x n f ( x ) {\displaystyle f^{(n)}(x)={\tfrac {d^{n}}{dx^{n}}}f(x)} . If one sets f ( x ) = e and g ( x ) = e , cancelling the common factor of e from each term gives the ordinary binomial theorem. For
11752-942: The theorem, the expansion of any nonnegative integer power n of the binomial x + y is a sum of the form ( x + y ) n = ( n 0 ) x n y 0 + ( n 1 ) x n − 1 y 1 + ( n 2 ) x n − 2 y 2 + ⋯ + ( n n − 1 ) x 1 y n − 1 + ( n n ) x 0 y n , {\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},} where each ( n k ) {\displaystyle {\tbinom {n}{k}}}
11865-400: The theory of elementary topoi , the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint . Cardinality In mathematics , cardinality describes a relationship between sets which compares their relative size. For example,
11978-543: The three 2-element subsets of {1, 2, 3} , namely, { 2 , 3 } , { 1 , 3 } , { 1 , 2 } , {\displaystyle \{2,3\},\;\{1,3\},\;\{1,2\},} where each subset specifies the positions of the y in a corresponding string. Induction yields another proof of the binomial theorem. When n = 0 , both sides equal 1 , since x = 1 and ( 0 0 ) = 1. {\displaystyle {\tbinom {0}{0}}=1.} Now suppose that
12091-1968: The usual double-angle identities. Similarly, since ( cos x + i sin x ) 3 = cos 3 x + 3 i cos 2 x sin x − 3 cos x sin 2 x − i sin 3 x , {\displaystyle \left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,} De Moivre's formula yields cos ( 3 x ) = cos 3 x − 3 cos x sin 2 x and sin ( 3 x ) = 3 cos 2 x sin x − sin 3 x . {\displaystyle \cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.} In general, cos ( n x ) = ∑ k even ( − 1 ) k / 2 ( n k ) cos n − k x sin k x {\displaystyle \cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x} and sin ( n x ) = ∑ k odd ( − 1 ) ( k − 1 ) / 2 ( n k ) cos n − k x sin k x . {\displaystyle \sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.} There are also similar formulas using Chebyshev polynomials . The number e
12204-513: The usual formula with factorials. However, for an arbitrary number r , one can define ( r k ) = r ( r − 1 ) ⋯ ( r − k + 1 ) k ! = ( r ) k k ! , {\displaystyle {r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},} where ( ⋅ ) k {\displaystyle (\cdot )_{k}}
12317-423: The vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize
12430-1081: The whole of any square, or cube, or hypercube , or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof . Cantor also showed that sets with cardinality strictly greater than c {\displaystyle {\mathfrak {c}}} exist (see his generalized diagonal argument and theorem ). They include, for instance: Both have cardinality The cardinal equalities c 2 = c , {\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},} c ℵ 0 = c , {\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},} and c c = 2 c {\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}} can be demonstrated using cardinal arithmetic : If A and B are disjoint sets , then From this, one can show that in general,
12543-454: The work of Henry Briggs ' Arithmetica Logarithmica (1624). Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of John Wallis 's Arithmetic Infinitorum and his method of interpolation. A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670. According to
12656-415: Was not defined as a specific object itself. However, such an object can be defined as follows. The relation of having the same cardinality is called equinumerosity , and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A . There are two ways to define the "cardinality of
12769-576: Was that the cardinality of the continuum ( c {\displaystyle {\mathfrak {c}}} ) is greater than that of the natural numbers ( ℵ 0 {\displaystyle \aleph _{0}} ); that is, there are more real numbers R than natural numbers N . Namely, Cantor showed that c = 2 ℵ 0 = ℶ 1 {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}} (see Beth one ) satisfies: The continuum hypothesis states that there
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