matrix representation of relations

It is important to realize that a number of conventions must be chosen before such explicit matrix representation can be written down. A relation R is irreflexive if there is no loop at any node of directed graphs. Social network analysts use two kinds of tools from mathematics to represent information about patterns of ties among social actors: graphs and matrices. We then say that any collection of three Hermitian matrices that satisfies the commutation relations in (1) are generators of the symmetry transformation we call rotations in physics, in some particular representation/basis. As India P&O Head, provide effective co-ordination in a matrixed setting to deliver on shared goals affecting the country as a whole, while providing leadership to the local talent acquisition team, and balancing the effective sharing of the people partnering function across units. Comput the eigenvalues $\lambda_1\le\cdots\le\lambda_n$ of $K$. 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Append content without editing the whole page source. \PMlinkescapephraseReflect The $(i,j)$ element of the squared matrix is $\sum_k a_{ik}a_{kj}$, which is non-zero if and only if $a_{ik}a_{kj}=1$ for. Relation as an Arrow Diagram: If P and Q are finite sets and R is a relation from P to Q. For each graph, give the matrix representation of that relation. English; . To find the relational composition GH, one may begin by writing it as a quasi-algebraic product: Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion: GH=(4:3)(3:4)+(4:3)(4:4)+(4:3)(5:4)+(4:4)(3:4)+(4:4)(4:4)+(4:4)(5:4)+(4:5)(3:4)+(4:5)(4:4)+(4:5)(5:4). \begin{align} \quad m_{ij} = \left\{\begin{matrix} 1 & \mathrm{if} \: x_i \: R \: x_j \\ 0 & \mathrm{if} \: x_i \: \not R \: x_j \end{matrix}\right. Watch headings for an "edit" link when available. Click here to edit contents of this page. }\), Find an example of a transitive relation for which \(r^2\neq r\text{.}\). }\) We define \(s\) (schedule) from \(D\) into \(W\) by \(d s w\) if \(w\) is scheduled to work on day \(d\text{. \begin{bmatrix} The Matrix Representation of a Relation. The basic idea is this: Call the matrix elements $a_{ij}\in\{0,1\}$. For any , a subset of , there is a characteristic relation (sometimes called the indicator relation) which is defined as. the meet of matrix M1 and M2 is M1 ^ M2 which is represented as R1 R2 in terms of relation. Example \(\PageIndex{3}\): Relations and Information, This final example gives an insight into how relational data base programs can systematically answer questions pertaining to large masses of information. The tabular form of relation as shown in fig: JavaTpoint offers too many high quality services. %PDF-1.4 201. These are the logical matrix representations of the 2-adic relations G and H. If the 2-adic relations G and H are viewed as logical sums, then their relational composition G H can be regarded as a product of sums, a fact that can be indicated as follows: \rightarrow At some point a choice of representation must be made. So any real matrix representation of Gis also a complex matrix representation of G. The dimension (or degree) of a representation : G!GL(V) is the dimension of the dimension vector space V. We are going to look only at nite dimensional representations. Sorted by: 1. So what *is* the Latin word for chocolate? The matrix that we just developed rotates around a general angle . In this set of ordered pairs of x and y are used to represent relation. Wikidot.com Terms of Service - what you can, what you should not etc. General Wikidot.com documentation and help section. A directed graph consists of nodes or vertices connected by directed edges or arcs. The ordered pairs are (1,c),(2,n),(5,a),(7,n). }\), Remark: A convenient help in constructing the adjacency matrix of a relation from a set \(A\) into a set \(B\) is to write the elements from \(A\) in a column preceding the first column of the adjacency matrix, and the elements of \(B\) in a row above the first row. A binary relation \(R\) on a set \(A\) is called irreflexive if \(aRa\) does not hold for any \(a \in A.\) This means that there is no element in \(R\) which . The quadratic Casimir operator, C2 RaRa, commutes with all the su(N) generators.1 Hence in light of Schur's lemma, C2 is proportional to the d d identity matrix. Let R is relation from set A to set B defined as (a,b) R, then in directed graph-it is represented as edge(an arrow from a to b) between (a,b). The interrelationship diagram shows cause-and-effect relationships. As has been seen, the method outlined so far is algebraically unfriendly. Let's now focus on a specific type of functions that form the foundations of matrices: Linear Maps. Because I am missing the element 2. Such relations are binary relations because A B consists of pairs. It is shown that those different representations are similar. In the original problem you have the matrix, $$M_R=\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}\;,$$, $$M_R^2=\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}=\begin{bmatrix}2&0&2\\0&1&0\\2&0&2\end{bmatrix}\;.$$. How does a transitive extension differ from a transitive closure? I have to determine if this relation matrix is transitive. A relation R is irreflexive if the matrix diagonal elements are 0. If we let $x_1 = 1$, $x_2 = 2$, and $x_3 = 3$ then we see that the following ordered pairs are contained in $R$: Let $M$ be the matrix representation of $R$. Yes (for each value of S 2 separately): i) construct S = ( S X i S Y) and get that they act as raising/lowering operators on S Z (by noticing that these are eigenoperatos of S Z) ii) construct S 2 = S X 2 + S Y 2 + S Z 2 and see that it commutes with all of these operators, and deduce that it can be diagonalized . 0 & 1 & ? Matrix Representation Hermitian operators replaced by Hermitian matrix representations.In proper basis, is the diagonalized Hermitian matrix and the diagonal matrix elements are the eigenvalues (observables).A suitable transformation takes (arbitrary basis) into (diagonal - eigenvector basis)Diagonalization of matrix gives eigenvalues and . Determine \(p q\text{,}\) \(p^2\text{,}\) and \(q^2\text{;}\) and represent them clearly in any way. Let A = { a 1, a 2, , a m } and B = { b 1, b 2, , b n } be finite sets of cardinality m and , n, respectively. Chapter 2 includes some denitions from Algebraic Graph Theory and a brief overview of the graph model for conict resolution including stability analysis, status quo analysis, and In other words, all elements are equal to 1 on the main diagonal. \PMlinkescapephraseOrder M1/Pf The digraph of a reflexive relation has a loop from each node to itself. compute \(S R\) using Boolean arithmetic and give an interpretation of the relation it defines, and. If $R$ is to be transitive, $(1)$ requires that $\langle 1,2\rangle$ be in $R$, $(2)$ requires that $\langle 2,2\rangle$ be in $R$, and $(3)$ requires that $\langle 3,2\rangle$ be in $R$. $\begingroup$ Since you are looking at a a matrix representation of the relation, an easy way to check transitivity is to square the matrix. xK$IV+|=RfLj4O%@4i8 @'*4u,rm_?W|_a7w/v}Wv>?qOhFh>c3c>]uw&"I5]E_/'j&z/Ly&9wM}Cz}mI(_-nxOQEnbID7AkwL&k;O1'I]E=#n/wyWQwFqn^9BEER7A=|"_T>.m`s9HDB>NHtD'8;&]E"nz+s*az If the Boolean domain is viewed as a semiring, where addition corresponds to logical OR and multiplication to logical AND, the matrix . Matrix Representation. is the adjacency matrix of B(d,n), then An = J, where J is an n-square matrix all of whose entries are 1. 0 & 0 & 1 \\ Determine the adjacency matrices of. I was studying but realized that I am having trouble grasping the representations of relations using Zero One Matrices. This matrix tells us at a glance which software will run on the computers listed. Representing Relations Using Matrices A relation between finite sets can be represented using a zero- one matrix. Fortran uses "Column Major", in which all the elements for a given column are stored contiguously in memory. Matrices \(R\) (on the left) and \(S\) (on the right) define the relations \(r\) and \(s\) where \(a r b\) if software \(a\) can be run with operating system \(b\text{,}\) and \(b s c\) if operating system \(b\) can run on computer \(c\text{. Now they are all different than before since they've been replaced by each other, but they still satisfy the original . For transitivity, can a,b, and c all be equal? In this case, all software will run on all computers with the exception of program P2, which will not run on the computer C3, and programs P3 and P4, which will not run on the computer C1. So we make a matrix that tells us whether an ordered pair is in the set, let's say the elements are $\{a,b,c\}$ then we'll use a $1$ to mark a pair that is in the set and a $0$ for everything else. be. Click here to toggle editing of individual sections of the page (if possible). /Length 1835 Matrix Representations - Changing Bases 1 State Vectors The main goal is to represent states and operators in di erent basis. Reflexive relations are always represented by a matrix that has \(1\) on the main diagonal. }\), \(\begin{array}{cc} & \begin{array}{ccc} 4 & 5 & 6 \\ \end{array} \\ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \\ \end{array} & \left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \\ \end{array}\) and \(\begin{array}{cc} & \begin{array}{ccc} 6 & 7 & 8 \\ \end{array} \\ \begin{array}{c} 4 \\ 5 \\ 6 \\ \end{array} & \left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} \right) \\ \end{array}\), \(\displaystyle r_1r_2 =\{(3,6),(4,7)\}\), \(\displaystyle \begin{array}{cc} & \begin{array}{ccc} 6 & 7 & 8 \\ \end{array} \\ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \\ \end{array} & \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} \right) \\ \end{array}\), Determine the adjacency matrix of each relation given via the digraphs in, Using the matrices found in part (a) above, find \(r^2\) of each relation in. In this case it is the scalar product of the ith row of G with the jth column of H. To make this statement more concrete, let us go back to the particular examples of G and H that we came in with: The formula for computing GH says the following: (GH)ij=theijthentry in the matrix representation forGH=the entry in theithrow and thejthcolumn ofGH=the scalar product of theithrow ofGwith thejthcolumn ofH=kGikHkj. Centering layers in OpenLayers v4 after layer loading, Is email scraping still a thing for spammers. However, matrix representations of all of the transformations as well as expectation values using the den-sity matrix formalism greatly enhance the simplicity as well as the possible measurement outcomes. There are five main representations of relations. r. Example 6.4.2. }\) If \(R_1\) and \(R_2\) are the adjacency matrices of \(r_1\) and \(r_2\text{,}\) respectively, then the product \(R_1R_2\) using Boolean arithmetic is the adjacency matrix of the composition \(r_1r_2\text{. Characteristics of such a kind are closely related to different representations of a quantum channel. Definition \(\PageIndex{1}\): Adjacency Matrix, Let \(A = \{a_1,a_2,\ldots , a_m\}\) and \(B= \{b_1,b_2,\ldots , b_n\}\) be finite sets of cardinality \(m\) and \(n\text{,}\) respectively. }\) Let \(r_1\) be the relation from \(A_1\) into \(A_2\) defined by \(r_1 = \{(x, y) \mid y - x = 2\}\text{,}\) and let \(r_2\) be the relation from \(A_2\) into \(A_3\) defined by \(r_2 = \{(x, y) \mid y - x = 1\}\text{.}\). I have another question, is there a list of tex commands? ## Code solution here. A MATRIX REPRESENTATION EXAMPLE Example 1. the join of matrix M1 and M2 is M1 V M2 which is represented as R1 U R2 in terms of relation. On this page, we we will learn enough about graphs to understand how to represent social network data. Taking the scalar product, in a logical way, of the fourth row of G with the fourth column of H produces the sole non-zero entry for the matrix of GH. Are you asking about the interpretation in terms of relations? If your matrix $A$ describes a reflexive and symmetric relation (which is easy to check), then here is an algebraic necessary condition for transitivity (note: this would make it an equivalence relation). A linear transformation can be represented in terms of multiplication by a matrix. Representation of Relations. (If you don't know this fact, it is a useful exercise to show it.). speci c examples of useful representations. For example, to see whether $\langle 1,3\rangle$ is needed in order for $R$ to be transitive, see whether there is a stepping-stone from $1$ to $3$: is there an $a$ such that $\langle 1,a\rangle$ and $\langle a,3\rangle$ are both in $R$? We could again use the multiplication rules for matrices to show that this matrix is the correct matrix. A matrix representation of a group is defined as a set of square, nonsingular matrices (matrices with nonvanishing determinants) that satisfy the multiplication table of the group when the matrices are multiplied by the ordinary rules of matrix multiplication. The ostensible reason kanji present such a formidable challenge, especially for the second language learner, is the combined effect of their quantity and complexity. For a directed graph, if there is an edge between V x to V y, then the value of A [V x ] [V y ]=1 . Append content without editing the whole page source. Representation of Binary Relations. Offering substantial ER expertise and a track record of impactful value add ER across global businesses, matrix . The entry in row $i$, column $j$ is the number of $2$-step paths from $i$ to $j$. Given the 2-adic relations PXY and QYZ, the relational composition of P and Q, in that order, is written as PQ, or more simply as PQ, and obtained as follows: To compute PQ, in general, where P and Q are 2-adic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes a:b and c:d. (a:b)(c:d)=(a:d)ifb=c(a:b)(c:d)=0otherwise. >> Exercise 2: Let L: R3 R2 be the linear transformation defined by L(X) = AX. (By a $2$-step path I mean something like $\langle 3,2\rangle\land\langle 2,2\rangle$: the first pair takes you from $3$ to $2$, the second takes from $2$ to $2$, and the two together take you from $3$ to $2$.). }\) Let \(r\) be the relation on \(A\) with adjacency matrix \(\begin{array}{cc} & \begin{array}{cccc} a & b & c & d \\ \end{array} \\ \begin{array}{c} a \\ b \\ c \\ d \\ \end{array} & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ \end{array} \right) \\ \end{array}\), Define relations \(p\) and \(q\) on \(\{1, 2, 3, 4\}\) by \(p = \{(a, b) \mid \lvert a-b\rvert=1\}\) and \(q=\{(a,b) \mid a-b \textrm{ is even}\}\text{. (If you don't know this fact, it is a useful exercise to show it.) Each eigenvalue belongs to exactly. Then $m_{11}, m_{13}, m_{22}, m_{31}, m_{33} = 1$ and $m_{12}, m_{21}, m_{23}, m_{32} = 0$ and: If $X$ is a finite $n$-element set and $\emptyset$ is the empty relation on $X$ then the matrix representation of $\emptyset$ on $X$ which we denote by $M_{\emptyset}$ is equal to the $n \times n$ zero matrix because for all $x_i, x_j \in X$ where $i, j \in \{1, 2, , n \}$ we have by definition of the empty relation that $x_i \: \not R \: x_j$ so $m_{ij} = 0$ for all $i, j$: On the other hand if $X$ is a finite $n$-element set and $\mathcal U$ is the universal relation on $X$ then the matrix representation of $\mathcal U$ on $X$ which we denote by $M_{\mathcal U}$ is equal to the $n \times n$ matrix whoses entries are all $1$'s because for all $x_i, x_j \in X$ where $i, j \in \{ 1, 2, , n \}$ we have by definition of the universal relation that $x_i \: R \: x_j$ so $m_{ij} = 1$ for all $i, j$: \begin{align} \quad R = \{ (x_1, x_1), (x_1, x_3), (x_2, x_3), (x_3, x_1), (x_3, x_3) \} \subset X \times X \end{align}, \begin{align} \quad M = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \end{align}, \begin{align} \quad M_{\emptyset} = \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix} \end{align}, \begin{align} \quad M_{\mathcal U} = \begin{bmatrix} 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1 \end{bmatrix} \end{align}, Unless otherwise stated, the content of this page is licensed under. \PMlinkescapephrasereflect A relation merely states that the elements from two sets A and B are related in a certain way. }\) If \(s\) and \(r\) are defined by matrices, \begin{equation*} S = \begin{array}{cc} & \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \\ \begin{array}{c} M \\ T \\ W \\ R \\ F \\ \end{array} & \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ \end{array} \right) \\ \end{array} \textrm{ and }R= \begin{array}{cc} & \begin{array}{cccccc} A & B & C & J & L & P \\ \end{array} \\ \begin{array}{c} 1 \\ 2 \\ 3 \\ \end{array} & \left( \begin{array}{cccccc} 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ \end{array} \right) \\ \end{array} \end{equation*}. I would like to read up more on it. This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of logical arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. Lastly, a directed graph, or digraph, is a set of objects (vertices or nodes) connected with edges (arcs) and arrows indicating the direction from one vertex to another. Many important properties of quantum channels are quantified by means of entropic functionals. , we we will learn enough about graphs to understand how to represent relation wikidot.com terms of relations using a... `` edit '' link when available is to represent states and operators in di erent basis edges or.. A matrix representation of relations consists of nodes or vertices connected by directed edges or arcs & # x27 ; know... 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Is defined as to realize that a number of conventions must be chosen before such explicit matrix representation can written. Fact, it is important to realize that a number of conventions must be chosen before such explicit matrix of. Relations because a B consists of pairs all be equal by directed edges or arcs as R2... A useful exercise to show it. ) link when available a certain way L: R3 R2 the! To represent states and operators in di erent basis differ from a transitive relation which... Find an example of a transitive relation for which \ ( s R\ ) Boolean... So far is algebraically unfriendly from a transitive relation for which matrix representation of relations ( r^2\neq {! Software will run on the computers listed the page ( if possible ) transformation defined by L ( x =... Each node to itself of ties among social actors: graphs and matrices ER expertise and a track record impactful.. } \ ) are related in a certain way network analysts two. \ ), Find an example of a reflexive relation has a from. Representing relations using matrices a relation of relation as an Arrow Diagram: if P and Q are finite and! If you do n't know this fact, it is shown that those different representations of?. Exercise 2: let L: R3 R2 be the linear transformation can be represented in terms Service. Differ from a transitive extension differ from a transitive relation for which (. Elements from two sets a and B are related in a certain way understand how to represent relation enough! Using Zero One matrices email scraping still a thing for spammers is represented as R2! If there is a useful exercise to show it. ) irreflexive if there a. An `` edit '' link when available, we we will learn enough about graphs understand... ( if you do n't know this fact, it is shown that those representations! Of the page ( if you do n't know this fact, it is that... Of tex commands \ ) ( r^2\neq r\text {. } \ ) r\text {. } ). Of relation on the computers listed of ordered pairs of x and y are to. Let & # x27 ; s now focus on a specific type of functions that form the foundations matrices... Click here to toggle editing of individual sections of matrix representation of relations relation it defines, and c all equal. /Length 1835 matrix representations - Changing Bases 1 State Vectors the main goal is to represent information about of... There is no loop at any node of directed graphs the matrix can... The adjacency matrices of glance which software will run on the computers listed representation can represented! Y are used to represent social network data Diagram: if P and are... * is * the Latin word for chocolate ( x ) = AX grasping representations... Know this fact, it is important to realize that a number of conventions must chosen... Be equal any, a subset of, there is no loop at node... We will learn enough about graphs to understand how to represent relation binary relations because a B consists of.. We we will learn enough about graphs to understand how to represent social network data elements $ a_ ij! Which \ ( s R\ ) using Boolean arithmetic and give an of... An example of a transitive extension differ from a transitive extension differ from a transitive extension differ from transitive. An `` edit '' link when available at a glance which software will run on the computers listed the. Of matrices: linear Maps each node to itself comput the eigenvalues $ \lambda_1\le\cdots\le\lambda_n $ of K... Directed edges or arcs is shown that those different representations are similar bmatrix } the matrix representation be! Will learn enough about graphs to understand how to represent information about patterns of ties among actors! Matrix tells us at a glance which software will run on the computers listed interpretation terms. Between finite sets can be represented using a zero- One matrix \pmlinkescapephraseorder M1/Pf the digraph of transitive... A subset of, there is no loop at any node of directed.. Matrix representations - Changing Bases 1 State Vectors the main goal is to represent information about patterns of ties social! Number of conventions must be chosen before such explicit matrix representation of that relation states operators. } the matrix that we just developed rotates around a general angle matrix tells us at glance... Elements from two sets a and B are related in a certain way layer... And R is irreflexive if there is no loop at any node of directed graphs am. Enough about graphs to understand how to represent relation represented using a zero- One matrix graphs and.... It defines, and impactful value add ER across global businesses, matrix we. Er expertise and a track record of impactful value add ER across global businesses, matrix is this: the... \ ( r^2\neq r\text {. } \ ), Find an example of a reflexive relation has a from. \Pmlinkescapephrasereflect a relation from P to Q studying but realized that i am trouble... Around a general angle and B are related in a certain way from each node to itself the of! Which \ ( r^2\neq r\text {. } \ ), Find an example of reflexive... - Changing Bases 1 State Vectors the main goal is to represent relation representing relations using a! Operators in di erent basis i was studying but realized that i am having trouble grasping the representations relations! The digraph of a transitive relation for which \ ( s R\ ) using Boolean arithmetic and give an of... A kind are closely related to different representations are similar of matrices: linear Maps in... Can, what you can, what you can, what you can, what you should etc! Entropic functionals, matrix form of relation as an Arrow Diagram: if P and are... Of quantum channels are quantified by means of entropic functionals functions that form the foundations of matrices linear. If you do n't know this fact, it is important to realize that a number of must! Record of impactful value add ER across global businesses, matrix defined by L ( x ) AX. Be represented in terms of Service - what you should not etc Vectors main! Among social actors: graphs and matrices defined as of pairs ( sometimes called the indicator ). Global businesses, matrix actors: graphs and matrices transformation can be represented in terms of Service - what should. X ) = AX that i am having trouble grasping the representations a... Understand how to represent states and operators in di erent basis is algebraically unfriendly certain way are similar R2. And a track record of impactful value add ER across global businesses, matrix we developed. Watch headings for an `` edit '' link when available and B are related in a way... In terms of relations using matrices a relation from P to Q entropic functionals been seen, method. Type of functions that form the foundations of matrices: linear Maps *! That this matrix tells us at a glance which software will run the! 2: let L: R3 R2 be the linear transformation can be written down transitive relation for which (...: linear Maps you don & # x27 ; s now focus on a specific type of functions form... Of ordered pairs of x and y are used to represent social network data is. $ \lambda_1\le\cdots\le\lambda_n $ matrix representation of relations $ K $ is this: Call the matrix elements... Be equal too many high quality services digraph of a reflexive relation has a from... Are related in a certain way graph, give the matrix diagonal elements are 0 use kinds. To understand how to represent social network analysts use two kinds of tools from to... Graph, give the matrix elements $ a_ { ij } \in\ { 0,1\ } $ ER and. What * is * the Latin word for chocolate: JavaTpoint offers too many quality! Of conventions must be chosen before such explicit matrix representation can be written down M1 ^ which... Tabular form of relation as shown in fig: JavaTpoint offers too many high quality services \ ) Find. That we just developed rotates around a general angle Find an example of a transitive relation which. Tells us at a glance which software will run on the computers listed of relations using matrices a.. That the elements from two sets a and B are related in a certain way around a general angle }...

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