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64-612: RPN may refer to: Science and technology [ edit ] Real projective space ( R P n , {\displaystyle \mathbb {R} \mathrm {P} ^{n},} ), a type of topological space Reverse Polish notation , a.k.a. postfix notation, a mathematical notation Registered Parameter Number , in MIDI Recherche en Prévision Numérique , weather forecasting research service, Canada Risk priority number in failure analysis such as FMEA, taking into account

128-432: A ∼ b {\displaystyle a\sim b} holds for all a {\displaystyle a} and b {\displaystyle b} in Y {\displaystyle Y} , and never for a {\displaystyle a} in Y {\displaystyle Y} and b {\displaystyle b} outside Y {\displaystyle Y} ,

192-523: A ≈ b {\displaystyle a\approx b} for all a , b ∈ S , {\displaystyle a,b\in S,} then ≈ {\displaystyle \approx } is said to be a coarser relation than ∼ {\displaystyle \sim } , and ∼ {\displaystyle \sim } is a finer relation than ≈ {\displaystyle \approx } . Equivalently, The equality equivalence relation

256-595: A , a ) , ( b , b ) , ( c , c ) , ( b , c ) , ( c , b ) } {\displaystyle R=\{(a,a),(b,b),(c,c),(b,c),(c,b)\}} is an equivalence relation. The following sets are equivalence classes of this relation: [ a ] = { a } ,         [ b ] = [ c ] = { b , c } . {\displaystyle [a]=\{a\},~~~~[b]=[c]=\{b,c\}.} The set of all equivalence classes for R {\displaystyle R}

320-463: A = c {\displaystyle a=c} (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes . Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Various notations are used in the literature to denote that two elements a {\displaystyle a} and b {\displaystyle b} of

384-451: A R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. In mathematics , an equivalence relation is a binary relation that is reflexive , symmetric and transitive . The equipollence relation between line segments in geometry

448-476: A ⁠ λ {\displaystyle \lambda } ⁠ such that ⁠ λ x {\displaystyle \lambda x} ⁠ has norm  1. There are precisely two such ⁠ λ {\displaystyle \lambda } ⁠ differing by sign. Thus ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ can also be formed by identifying antipodal points of

512-413: A geometric lattice . Much of mathematics is grounded in the study of equivalences, and order relations . Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to

576-402: A ≡ R b ", or " a R ⁡ b {\displaystyle {a\mathop {R} b}} " to specify R {\displaystyle R} explicitly. Non-equivalence may be written " a ≁ b " or " a ≢ b {\displaystyle a\not \equiv b} ". A binary relation ∼ {\displaystyle \,\sim \,} on

640-624: A commutative triangle. See also invariant . Some authors use "compatible with ∼ {\displaystyle \,\sim } " or just "respects ∼ {\displaystyle \,\sim } " instead of "invariant under ∼ {\displaystyle \,\sim } ". More generally, a function may map equivalent arguments (under an equivalence relation ∼ A {\displaystyle \,\sim _{A}} ) to equivalent values (under an equivalence relation ∼ B {\displaystyle \,\sim _{B}} ). Such

704-458: A constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry). For the standard round metric, this has sectional curvature identically 1. In the standard round metric, the measure of projective space is exactly half the measure of the sphere. Real projective spaces are smooth manifolds . On S , in homogeneous coordinates, ( x 1 , ..., x n +1 ), consider

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768-449: A function f {\displaystyle f} is the equivalence relation ~ defined by x ∼ y  if and only if  f ( x ) = f ( y ) . {\displaystyle x\sim y{\text{ if and only if }}f(x)=f(y).} The equivalence kernel of an injection is the identity relation . A partition of X is a set P of nonempty subsets of X , such that every element of X

832-588: A function is known as a morphism from ∼ A {\displaystyle \,\sim _{A}} to ∼ B . {\displaystyle \,\sim _{B}.} Let a , b ∈ X {\displaystyle a,b\in X} , and ∼ {\displaystyle \sim } be an equivalence relation. Some key definitions and terminology follow: A subset Y {\displaystyle Y} of X {\displaystyle X} such that

896-506: A lesser extent, on the theory of lattices, categories , and groupoids . Just as order relations are grounded in ordered sets , sets closed under pairwise supremum and infimum , equivalence relations are grounded in partitioned sets , which are sets closed under bijections that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations . Hence permutation groups (also known as transformation groups ) and

960-469: A set X {\displaystyle X} is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a , b , {\displaystyle a,b,} and c {\displaystyle c} in X : {\displaystyle X:} X {\displaystyle X} together with the relation ∼ {\displaystyle \,\sim \,}

1024-412: A set are equivalent with respect to an equivalence relation R ; {\displaystyle R;} the most common are " a ∼ b {\displaystyle a\sim b} " and " a ≡ b ", which are used when R {\displaystyle R} is implicit, and variations of " a ∼ R b {\displaystyle a\sim _{R}b} ", "

1088-419: A unique morphism from x to y if and only if x ∼ y . {\displaystyle x\sim y.} The advantages of regarding an equivalence relation as a special case of a groupoid include: The equivalence relations on any set X , when ordered by set inclusion , form a complete lattice , called Con X by convention. The canonical map ker  : X ^ X → Con X , relates

1152-422: Is H ∗ ( R P ∞ ; Z / 2 Z ) = Z / 2 Z [ w 1 ] , {\displaystyle H^{*}(\mathbf {RP} ^{\infty };\mathbf {Z} /2\mathbf {Z} )=\mathbf {Z} /2\mathbf {Z} [w_{1}],} where w 1 {\displaystyle w_{1}} is the first Stiefel–Whitney class : it

1216-460: Is { { a } , { b , c } } . {\displaystyle \{\{a\},\{b,c\}\}.} This set is a partition of the set X {\displaystyle X} . It is also called the quotient set of X {\displaystyle X} by R {\displaystyle R} . The following relations are all equivalence relations: If ∼ {\displaystyle \,\sim \,}

1280-447: Is ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ . This action is actually a covering space action giving ⁠ S n {\displaystyle S^{n}} ⁠ as a double cover of ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ . Since ⁠ S n {\displaystyle S^{n}} ⁠

1344-550: Is simply connected for ⁠ n ≥ 2 {\displaystyle n\geq 2} ⁠ , it also serves as the universal cover in these cases. It follows that the fundamental group of ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ is ⁠ Z 2 {\displaystyle \mathbb {Z} _{2}} ⁠ when ⁠ n > 1 {\displaystyle n>1} ⁠ . (When n = 1 {\displaystyle n=1}

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1408-772: Is a topological space , there is a natural way of transforming X / ∼ {\displaystyle X/\sim } into a topological space; see Quotient space for the details. The projection of ∼ {\displaystyle \,\sim \,} is the function π : X → X / ∼ {\displaystyle \pi :X\to X/{\mathord {\sim }}} defined by π ( x ) = [ x ] {\displaystyle \pi (x)=[x]} which maps elements of X {\displaystyle X} into their respective equivalence classes by ∼ . {\displaystyle \,\sim .} The equivalence kernel of

1472-461: Is a CW complex with 1 cell in every dimension up to n . The cells are Schubert cells , as on the flag manifold . That is, take a complete flag (say the standard flag) 0 = V 0 < V 1 <...< V n ; then the closed k -cell is lines that lie in V k . Also the open k -cell (the interior of the k -cell) is lines in V k \ V k −1 (lines in V k but not V k −1 ). In homogeneous coordinates (with respect to

1536-434: Is a common example of an equivalence relation. A simpler example is equality. Any number a {\displaystyle a} is equal to itself (reflexive). If a = b {\displaystyle a=b} , then b = a {\displaystyle b=a} (symmetric). If a = b {\displaystyle a=b} and b = c {\displaystyle b=c} , then

1600-406: Is a generalisation of the definition of functional composition . The defining properties of an equivalence relation R {\displaystyle R} on a set X {\displaystyle X} can then be reformulated as follows: On the set X = { a , b , c } {\displaystyle X=\{a,b,c\}} , the relation R = { (

1664-417: Is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X . If ∼ {\displaystyle \sim } and ≈ {\displaystyle \approx } are two equivalence relations on the same set S {\displaystyle S} , and a ∼ b {\displaystyle a\sim b} implies

1728-399: Is an easy example of a theory which is ω- categorical , but not categorical for any larger cardinal number . An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all

1792-407: Is an element of a single element of P . Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X . Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X , and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X , which

1856-456: Is an equivalence relation on X , {\displaystyle X,} and P ( x ) {\displaystyle P(x)} is a property of elements of X , {\displaystyle X,} such that whenever x ∼ y , {\displaystyle x\sim y,} P ( x ) {\displaystyle P(x)} is true if P ( y ) {\displaystyle P(y)}

1920-711: Is called a setoid . The equivalence class of a {\displaystyle a} under ∼ , {\displaystyle \,\sim ,} denoted [ a ] , {\displaystyle [a],} is defined as [ a ] = { x ∈ X : x ∼ a } . {\displaystyle [a]=\{x\in X:x\sim a\}.} In relational algebra , if R ⊆ X × Y {\displaystyle R\subseteq X\times Y} and S ⊆ Y × Z {\displaystyle S\subseteq Y\times Z} are relations, then

1984-543: Is called an equivalence class of X {\displaystyle X} by ∼ {\displaystyle \sim } . Let [ a ] := { x ∈ X : a ∼ x } {\displaystyle [a]:=\{x\in X:a\sim x\}} denote the equivalence class to which a {\displaystyle a} belongs. All elements of X {\displaystyle X} equivalent to each other are also elements of

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2048-512: Is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the ⁠ n {\displaystyle n} ⁠ -sphere, a simply connected space. It is a double cover . The antipode map on ⁠ R p {\displaystyle \mathbb {R} ^{p}} ⁠ has sign ( − 1 ) p {\displaystyle (-1)^{p}} , so it

2112-442: Is different from Wikidata All article disambiguation pages All disambiguation pages Real projective space As with all projective spaces , ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ is formed by taking the quotient of R n + 1 ∖ { 0 } {\displaystyle \mathbb {R} ^{n+1}\setminus \{0\}} under

2176-798: Is not a regular CW structure, as the attaching maps are 2-to-1. However, its cover is a regular CW structure on the sphere, with 2 cells in every dimension; indeed, the minimal regular CW structure on the sphere. In light of the smooth structure, the existence of a Morse function would show RP is a CW complex. One such function is given by, in homogeneous coordinates, g ( x 1 , … , x n + 1 ) = ∑ i = 1 n + 1 i ⋅ | x i | 2 . {\displaystyle g(x_{1},\ldots ,x_{n+1})=\sum _{i=1}^{n+1}i\cdot |x_{i}|^{2}.} On each neighborhood U i , g has nondegenerate critical point (0,...,1,...,0) where 1 occurs in

2240-706: Is orientable if and only if ⁠ n + 1 {\displaystyle n+1} ⁠ is even, i.e., ⁠ n {\displaystyle n} ⁠ is odd. The projective ⁠ n {\displaystyle n} ⁠ -space is in fact diffeomorphic to the submanifold of R ( n + 1 ) 2 {\displaystyle \mathbb {R} ^{(n+1)^{2}}} consisting of all symmetric ⁠ ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} ⁠ matrices of trace 1 that are also idempotent linear transformations. Real projective space admits

2304-426: Is orientable if and only if n is odd, as the above homology calculation shows. The infinite real projective space is constructed as the direct limit or union of the finite projective spaces: R P ∞ := lim n R P n . {\displaystyle \mathbf {RP} ^{\infty }:=\lim _{n}\mathbf {RP} ^{n}.} This space is classifying space of O (1) ,

2368-529: Is orientation-preserving if and only if ⁠ p {\displaystyle p} ⁠ is even. The orientation character is thus: the non-trivial loop in π 1 ( R P n ) {\displaystyle \pi _{1}(\mathbb {RP} ^{n})} acts as ( − 1 ) n + 1 {\displaystyle (-1)^{n+1}} on orientation, so ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠

2432-454: Is said to be a morphism for ∼ , {\displaystyle \,\sim ,} a class invariant under ∼ , {\displaystyle \,\sim ,} or simply invariant under ∼ . {\displaystyle \,\sim .} This occurs, e.g. in the character theory of finite groups. The latter case with the function f {\displaystyle f} can be expressed by

2496-435: Is the n th Bell number B n : A key result links equivalence relations and partitions: In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongs to a unique cell of any partition of X , and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there

2560-405: Is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. The relation " ∼ {\displaystyle \sim } is finer than ≈ {\displaystyle \approx } " on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection

2624-445: Is the free Z / 2 Z {\displaystyle \mathbf {Z} /2\mathbf {Z} } -algebra on w 1 {\displaystyle w_{1}} , which has degree 1. Equivalence relation All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if

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2688-749: Is true, then the property P {\displaystyle P} is said to be well-defined or a class invariant under the relation ∼ . {\displaystyle \,\sim .} A frequent particular case occurs when f {\displaystyle f} is a function from X {\displaystyle X} to another set Y ; {\displaystyle Y;} if x 1 ∼ x 2 {\displaystyle x_{1}\sim x_{2}} implies f ( x 1 ) = f ( x 2 ) {\displaystyle f\left(x_{1}\right)=f\left(x_{2}\right)} then f {\displaystyle f}

2752-404: The ⁠ n {\displaystyle n} ⁠ -sphere (the map sending ⁠ x {\displaystyle x} ⁠ to ⁠ − x {\displaystyle -x} ⁠ ) generates a Z 2 group action on ⁠ S n {\displaystyle S^{n}} ⁠ . As mentioned above, the orbit space for this action

2816-525: The composite relation S R ⊆ X × Z {\displaystyle SR\subseteq X\times Z} is defined so that x S R z {\displaystyle x\,SR\,z} if and only if there is a y ∈ Y {\displaystyle y\in Y} such that x R y {\displaystyle x\,R\,y} and y S z {\displaystyle y\,S\,z} . This definition

2880-484: The equivalence relation ⁠ x ∼ λ x {\displaystyle x\sim \lambda x} ⁠ for all real numbers ⁠ λ ≠ 0 {\displaystyle \lambda \neq 0} ⁠ . For all ⁠ x {\displaystyle x} ⁠ in R n + 1 ∖ { 0 } {\displaystyle \mathbb {R} ^{n+1}\setminus \{0\}} one can always find

2944-399: The i -th position with Morse index i . This shows RP is a CW complex with 1 cell in every dimension. Real projective space has a natural line bundle over it, called the tautological bundle . More precisely, this is called the tautological subbundle, and there is also a dual n -dimensional bundle called the tautological quotient bundle. The higher homotopy groups of RP are exactly

3008-444: The monoid X ^ X of all functions on X and Con X . ker is surjective but not injective . Less formally, the equivalence relation ker on X , takes each function f  : X → X to its kernel ker f . Likewise, ker(ker) is an equivalence relation on X ^ X . Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes

3072-471: The above CW structure has 1 cell in each dimension 0, ..., n . For each dimensional k , the boundary maps d k  : δ D → RP / RP is the map that collapses the equator on S and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): deg ⁡ ( d k ) = 1 + ( − 1 ) k . {\displaystyle \deg(d_{k})=1+(-1)^{k}.} Thus

3136-505: The arguments of the transformation group operations composition and inverse are elements of a set of bijections , A → A . Moving to groups in general, let H be a subgroup of some group G . Let ~ be an equivalence relation on G , such that a ∼ b  if and only if  a b − 1 ∈ H . {\displaystyle a\sim b{\text{ if and only if }}ab^{-1}\in H.} The equivalence classes of ~—also called

3200-539: The bounding equator. This shows that ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ is also equivalent to the closed ⁠ n {\displaystyle n} ⁠ -dimensional disk, ⁠ D n {\displaystyle D^{n}} ⁠ , with antipodal points on the boundary, ∂ D n = S n − 1 {\displaystyle \partial D^{n}=S^{n-1}} , identified. The antipodal map on

3264-550: The coordinate neighborhood U 1 = {( x 1 ... x n +1 ) | x 1 ≠ 0} can be identified with the interior of n -disk D . When x i = 0, one has RP . Therefore the n −1 skeleton of RP is RP , and the attaching map f  : S → RP is the 2-to-1 covering map. One can put R P n = R P n − 1 ∪ f D n . {\displaystyle \mathbf {RP} ^{n}=\mathbf {RP} ^{n-1}\cup _{f}D^{n}.} Induction shows that RP

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3328-659: The first orthogonal group . The double cover of this space is the infinite sphere S ∞ {\displaystyle S^{\infty }} , which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space K ( Z 2 , 1). For each nonnegative integer q , the modulo 2 homology group H q ( R P ∞ ; Z / 2 ) = Z / 2 {\displaystyle H_{q}(\mathbf {RP} ^{\infty };\mathbf {Z} /2)=\mathbf {Z} /2} . Its cohomology ring modulo 2

3392-476: The flag), the cells are [ ∗ : 0 : 0 : ⋯ : 0 ] [ ∗ : ∗ : 0 : ⋯ : 0 ] ⋮ [ ∗ : ∗ : ∗ : ⋯ : ∗ ] . {\displaystyle {\begin{array}{c}[*:0:0:\dots :0]\\{[}*:*:0:\dots :0]\\\vdots \\{[}*:*:*:\dots :*].\end{array}}} This

3456-462: The following three connected theorems hold: In sum, given an equivalence relation ~ over A , there exists a transformation group G over A whose orbits are the equivalence classes of A under ~. This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe A . Meanwhile,

3520-609: The fundamental group is ⁠ Z {\displaystyle \mathbb {Z} } ⁠ due to the homeomorphism with ⁠ S 1 {\displaystyle S^{1}} ⁠ ). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in ⁠ S n {\displaystyle S^{n}} ⁠ down to ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ . The projective ⁠ n {\displaystyle n} ⁠ -space

3584-1373: The higher homotopy groups of S , via the long exact sequence on homotopy associated to a fibration . Explicitly, the fiber bundle is: Z 2 → S n → R P n . {\displaystyle \mathbf {Z} _{2}\to S^{n}\to \mathbf {RP} ^{n}.} You might also write this as S 0 → S n → R P n {\displaystyle S^{0}\to S^{n}\to \mathbf {RP} ^{n}} or O ( 1 ) → S n → R P n {\displaystyle O(1)\to S^{n}\to \mathbf {RP} ^{n}} by analogy with complex projective space . The homotopy groups are: π i ( R P n ) = { 0 i = 0 Z i = 1 , n = 1 Z / 2 Z i = 1 , n > 1 π i ( S n ) i > 1 , n > 0. {\displaystyle \pi _{i}(\mathbf {RP} ^{n})={\begin{cases}0&i=0\\\mathbf {Z} &i=1,n=1\\\mathbf {Z} /2\mathbf {Z} &i=1,n>1\\\pi _{i}(S^{n})&i>1,n>0.\end{cases}}} The cellular chain complex associated to

3648-576: The integral homology is H i ( R P n ) = { Z i = 0  or  i = n  odd, Z / 2 Z 0 < i < n ,   i   odd, 0 else. {\displaystyle H_{i}(\mathbf {RP} ^{n})={\begin{cases}\mathbf {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbf {Z} /2\mathbf {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{else.}}\end{cases}}} RP

3712-491: The orbits of the action of H on G —are the right cosets of H in G . Interchanging a and b yields the left cosets. Related thinking can be found in Rosen (2008: chpt. 10). Let G be a set and let "~" denote an equivalence relation over G . Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G , and for any two elements x and y of G , there exists

3776-590: The related notion of orbit shed light on the mathematical structure of equivalence relations. Let '~' denote an equivalence relation over some nonempty set A , called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A , meaning that for all x ∈ A {\displaystyle x\in A} and g ∈ G , g ( x ) ∈ [ x ] . {\displaystyle g\in G,g(x)\in [x].} Then

3840-587: The same equivalence class. The set of all equivalence classes of X {\displaystyle X} by ∼ , {\displaystyle \sim ,} denoted X / ∼ := { [ x ] : x ∈ X } , {\displaystyle X/{\mathord {\sim }}:=\{[x]:x\in X\},} is the quotient set of X {\displaystyle X} by ∼ . {\displaystyle \sim .} If X {\displaystyle X}

3904-464: The same term [REDACTED] This disambiguation page lists articles associated with the title RPN . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=RPN&oldid=1232047458 " Category : Disambiguation pages Hidden categories: Articles containing Malay (macrolanguage)-language text Short description

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3968-516: The severity, probability and detection probability of a failure event Nursing [ edit ] Registered practical nurse , nursing title Registered psychiatric nurse , nursing title specialising mental health Other uses [ edit ] Radio Philippines Network , Channel 9 Rosh Pina Airport (IATA code), Israel Rancangan Perumahan Negara (National Housing Programme), public housing in Brunei Topics referred to by

4032-417: The subset U i with x i ≠ 0. Each U i is homeomorphic to the disjoint union of two open unit balls in R that map to the same subset of RP and the coordinate transition functions are smooth. This gives RP a smooth structure . Real projective space RP admits the structure of a CW complex with 1 cell in every dimension. In homogeneous coordinates ( x 1 ... x n +1 ) on S ,

4096-427: The unit ⁠ n {\displaystyle n} ⁠ - sphere , ⁠ S n {\displaystyle S^{n}} ⁠ , in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} . One can further restrict to the upper hemisphere of ⁠ S n {\displaystyle S^{n}} ⁠ and merely identify antipodal points on

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