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4.1: Prelude to Systems of Distinct Representatives - Mathematics


Suppose that the student clubs at a college each send a representative to the student government from among the members of the club. So the first substantive question is: is there anything useful or interesting we can say about under what conditions it is possible to choose such representatives.

We turn this into a more mathematical situation:

Definition: system of distinct representatives (SDR)

Suppose that (A_1,A_2,ldots,A_n) are sets, which we refer to as a set system. A (complete) system of distinct representatives is a set ({x_1, x_2, ldots x_n}) such that (x_iin A_i) for all (i), and no two of the (x_i) are the same. A (partial) system of distinct representatives is a set of distinct elements ({x_1, x_2, ldots x_k}) such that (x_iin A_{j_i}), where (j_1,j_2,ldots,j_k) are distinct integers in ([n]).

In standard usage, "system of distinct representatives'' means "complete system of distinct representatives'', but it will be convenient to let "system of distinct representatives'' mean either a complete or partial system of distinct representatives depending on context. We usually abbreviate "system of distinct representatives'' as sdr.

We will analyze this problem in two ways: combinatorially and using graph theory.


Arrow's impossibility theorem

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare". [1]

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no "dictator": no single voter possesses the power to always determine the group's preference.

Cardinal voting electoral systems are not covered by the theorem, as they convey more information than rank orders. [2] [3] However, Gibbard's theorem shows that strategic voting remains a problem.

The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem. [4]

The practical consequences of the theorem are debatable: Arrow has said "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times." [5]


Solution

The three points and the segments joining them are plotted and labeled below:

To calculate the distance between two of the points, $A$ and $B$ for example, we can use the Pythagorean Theorem provided we can find a right triangle which has $overline$ as one side. The vertical and horizontal grid lines are perpendicular to one another so we can make a right angle by choosing one vertical grid line segment and one horizontal grid line segment as the legs of our triangle. This is also pictured above as $ riangle ADB$ is a right triangle with right angle $D$. We can apply the Pythagorean theorem to $ riangle ADB$ to find $|AB|^2 = |AD|^2 + |BD|^2.$We know $|AD| = 2$ units and $|BD| = 6$ units by counting squares on the coordinate grid. So this means $|AD| = sqrt<40>$ units. Applying the same technique to $ riangle AEC$ with right angle $E$ we find $|AC| = sqrt <25>= 5$ units. Using $ riangle CFB$ we find that $|CB| = sqrt<45>$ units.

Looking at part (b), we need to find the horizontal and vertical distance covered to go from $(u,v)$ to $(s,t)$. Moving from $u$ to $s$ requires a horizontal displacement of $s-u$ units while going from $v$ to $t$ is a vertical displacement of $t-v$ units. Applying the pattern in part (b), the distance from $(u,v)$ to $(s,t)$ is $sqrt<(s-u)^2 + (t-v)^2>$.

We can verify this formula and pattern by drawing a representative picture as in part (a), though we have not labeled the numbers on the axes because we do not know the exact coordinates of the points:


There are many other possible pictures as $(u,v)$ or $(s,t)$ could lie on one of the coordinate axes or could be in different quadrants. The Pythagorean relationship, however, holds regardless of where we translate the image.

In order to find the distance from $(u,v)$ to $(s,t)$ we can make these points two vertices of a rectangle, with horizontal and vertical sides, as drawn above. We let $A = (s,t)$ and $B = (u,v)$ in the following calculations. Since side $overline$ is vertical, this means that the $x$-coordinate of $Q$ is the same as that of $B=(u,v)$, namely $u$. Similarly, the $y$-coordinate of $Q$ is the same as the $y$-coordinate of $A = (s,t)$, namely $t$. So $Q = (u,t)$. Leg $overline$ has length $|t-v|$ while leg $QA$ has length $|s-u|$. Applying the Pythagorean theorem we find that the distance from $(u,v)$ to $(s,t)$ is $sqrt<|s-u|^2 + |t-v|^2>$. This is the same as what we found above: the square of any number is non-negative so $|s-u|^2 = (s-u)^2$ and $|t-v|^2 = (t-v)^2$.


The permanent of an n-by-n matrix A = (ai,j) is defined as

The sum here extends over all elements σ of the symmetric group Sn i.e. over all permutations of the numbers 1, 2, . n.

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account.

The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix. [2] Muir and Metzler use the notation | + | + <|>>quad <|>>> . [3]

The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function, [4] and was used by Muir and Metzler [5] in the modern, more specific, sense. [6]

If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). Furthermore, given a square matrix A = ( a i j ) ight)> of order n: [7]

  • perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may be written symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matricesP and Q,
  • multiplying any single row or column of A by a scalars changes perm(A) to s⋅perm(A),
  • perm(A) is invariant under transposition, that is, perm(A) = perm(A T ).

On the other hand, the basic multiplicative property of determinants is not valid for permanents. [9] A simple example shows that this is so.

A formula similar to Laplace's for the development of a determinant along a row, column or diagonal is also valid for the permanent [10] all signs have to be ignored for the permanent. For example, expanding along the first column,

while expanding along the last row gives,

Unlike the determinant, the permanent has no easy geometrical interpretation it is mainly used in combinatorics, in treating boson Green's functions in quantum field theory, and in determining state probabilities of boson sampling systems. [11] However, it has two graph-theoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite graph.

Symmetric tensors Edit

Cycle covers Edit

If the weight of a cycle-cover is defined to be the product of the weights of the arcs in each cycle, then

Perfect matchings Edit

Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph.

Enumeration Edit

The answers to many counting questions can be computed as permanents of matrices that only have 0 and 1 as entries.

Let Ω(n,k) be the class of all (0, 1)-matrices of order n with each row and column sum equal to k. Every matrix A in this class has perm(A) > 0. [13] The incidence matrices of projective planes are in the class Ω(n 2 + n + 1, n + 1) for n an integer > 1. The permanents corresponding to the smallest projective planes have been calculated. For n = 2, 3, and 4 the values are 24, 3852 and 18,534,400 respectively. [13] Let Z be the incidence matrix of the projective plane with n = 2, the Fano plane. Remarkably, perm(Z) = 24 = |det (Z)|, the absolute value of the determinant of Z. This is a consequence of Z being a circulant matrix and the theorem: [14]

If A is a circulant matrix in the class Ω(n,k) then if k > 3, perm(A) > |det (A)| and if k = 3, perm(A) = |det (A)|. Furthermore, when k = 3, by permuting rows and columns, A can be put into the form of a direct sum of e copies of the matrix Z and consequently, n = 7e and perm(A) = 24 e .

Permanents can also be used to calculate the number of permutations with restricted (prohibited) positions. For the standard n-set <1, 2, . n>, let A = ( a i j ) )> be the (0, 1)-matrix where aij = 1 if ij is allowed in a permutation and aij = 0 otherwise. Then perm(A) is equal to the number of permutations of the n-set that satisfy all the restrictions. [10] Two well known special cases of this are the solution of the derangement problem and the ménage problem: the number of permutations of an n-set with no fixed points (derangements) is given by

where J is the n×n all 1's matrix and I is the identity matrix, and the ménage numbers are given by

where I' is the (0, 1)-matrix with nonzero entries in positions (i, i + 1) and (n, 1).

Bounds Edit

The Bregman–Minc inequality, conjectured by H. Minc in 1963 [15] and proved by L. M. Brégman in 1973, [16] gives an upper bound for the permanent of an n × n (0, 1)-matrix. If A has ri ones in row i for each 1 ≤ in, the inequality states that

In 1926 Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/n n , achieved by the matrix for which all entries are equal to 1/n. [17] Proofs of this conjecture were published in 1980 by B. Gyires [18] and in 1981 by G. P. Egorychev [19] and D. I. Falikman [20] Egorychev's proof is an application of the Alexandrov–Fenchel inequality. [21] For this work, Egorychev and Falikman won the Fulkerson Prize in 1982. [22]

It may be rewritten in terms of the matrix entries as follows:

The permanent is believed to be more difficult to compute than the determinant. While the determinant can be computed in polynomial time by Gaussian elimination, Gaussian elimination cannot be used to compute the permanent. Moreover, computing the permanent of a (0,1)-matrix is #P-complete. Thus, if the permanent can be computed in polynomial time by any method, then FP = #P, which is an even stronger statement than P = NP. When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of ε M , where M is the value of the permanent and ε > 0 is arbitrary. [25] The permanent of a certain set of positive semidefinite matrices can also be approximated in probabilistic polynomial time: the best achievable error of this approximation is ε M >> ( M is again the value of the permanent). [26]

As a generalization, for any sequence of n non-negative integers, s 1 , s 2 , … , s n ,s_<2>,dots ,s_> define:

MacMahon's Master Theorem relating permanents and determinants is: [28]

where I is the order n identity matrix and X is the diagonal matrix with diagonal [ x 1 , x 2 , … , x n ] . ,x_<2>,dots ,x_].>

The permanent function can be generalized to apply to non-square matrices. Indeed, several authors make this the definition of a permanent and consider the restriction to square matrices a special case. [29] Specifically, for an m × n matrix A = ( a i j ) )> with mn, define

where P(n,m) is the set of all m-permutations of the n-set <1,2. n>. [30]

Systems of distinct representatives Edit

The generalization of the definition of a permanent to non-square matrices allows the concept to be used in a more natural way in some applications. For instance:

Let S1, S2, . Sm be subsets (not necessarily distinct) of an n-set with mn. The incidence matrix of this collection of subsets is an m × n (0,1)-matrix A. The number of systems of distinct representatives (SDR's) of this collection is perm(A). [31]


4. Diagrammatic Systems in Geometry

Mathematicians have used, and continue to use, diagrams extensively. The communication of mathematical concepts and proofs&mdashin textbooks, on blackboards&mdashis not uniformly sentential. Figures and pictures are common. In line with the prevailing conception of logic as essentially sentential, however, they are not usually thought to play a role in rigorous mathematical reasoning. Their use is taken to be limited to enhancing comprehension of a proof. They are not standardly believed to form any part of the proof itself.

The attitude is well illustrated by the standard assessment of Euclid&rsquos methodology in the Elements. In no mathematical subject are diagrams more prominent than in the elementary geometry Euclid develops in the text. The proofs of the subject seem to be in some sense about the diagrams of triangles and circles that appear with them. This is especially the case with the geometric proofs of the Elements. Diagrams for Euclid are not merely illustrative. Some of his inference steps depend on an appropriately constructed diagram. On the standard story, these steps indicate gaps in Euclid&rsquos proofs. They show how Euclid did not fully carry out the project of developing geometry axiomatically.

Ken Manders set out to explode this story with his seminal work &ldquoThe Euclidean diagram&rdquo (2008 [1995]). His analysis of Euclid&rsquos diagrammatic proof method reveals that Euclid employs diagrams in a controlled, systematic way. It thus calls into question the common, negative assessment of the rigor of the Elements. Moreover, the specifics of Manders&rsquo analysis suggest that the proofs of the text can be understood to adhere to a formal diagrammatic logic. This was subsequently confirmed by the development of formal diagrammatic systems designed to characterize such a logic. The first of these was FG (presented in Miller 2007), followed by the system Eu (Mumma 2010).

This section is devoted to explicating Manders&rsquo analysis and the formal systems that have emerged from it. After a brief survey of how Euclid&rsquos diagrams have been viewed through the centuries, Manders&rsquo picture of their role in geometric proofs is presented. A description of how the systems FG and Eu render this picture in formal terms and characterize a logic of Euclidean diagrams then follows.

4.1 Views on Euclid&rsquos diagrams from 4 th century BCE to the 20 th century CE

The elementary geometry of the Elements was taken to be foundational to mathematics from its inception in ancient Greece until the 19 th century. Accordingly, philosophers concerned with the nature of mathematics found themselves obliged to comment on the diagrammatic proofs of the text. A central issue, if not the central issue, was the generality problem. The diagram that appears with a Euclidean proof provides a single instantiation of the type of geometric configurations the proof is about. Yet properties seen to hold in the diagram are taken to hold of all the configurations of the given type. What justifies this jump from the particular to the general?

As an illustration, consider the proof for proposition 16 of book I of the Elements.

  • Let ABC be a triangle, and let one side of it BC be produced to D
  • I say that the angle ACD is greater than the interior and opposite angle BAC.
  • Let AC be bisected at E [I, 10], and let BE be joined and produced in a straight line to F
  • let EF be made equal to BE [I,3], and let FC be joined.
  • Then, since AE is equal to EC, and BE equal to EF, the two sides AE, EB are equal to the two sides CE, EF respectively and the angle AEB is equal to the angle FEC [I, 15].
  • Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE [I,4]therefore the angle BAE is equal to the angle ECF (which is also the angle ACF)
  • But the angle ACD is greater than the angle ACF
  • Therefore the angle ACD is greater than BAE.

The proof seems to refer to the parts of the diagram given with the proof. Nevertheless the proof does not purport to establish something just about the triangle in the diagram, but something about all triangles. The diagram thus serves to represent, in some way, all triangles.

The role of diagrams as representations is remarked upon by Aristotle in book A, chapter 10 of the Posterior Analytics:

The geometer bases no conclusion on the particular line he has drawn being that which he has described, but [refers to] what is illustrated by the figures. (The translation is by T. Heath, found in Euclid 1956: vol. I, p.119)

Aristotle does not in passage confront the question of how the geometer uses diagrams to reason about what they illustrate. A few centuries later Proclus does in his commentary on the Elements. Proclus asserts that passing from a particular instance to a universal conclusion is justified because geometers

&hellip use the objects set out in the diagram not as these particular figures, but as figures resembling others of the same sort. It is not as having such-and-such a size that the angle before me is bisected, but as being rectilinear and nothing more&hellipSuppose the given angle is a right angle&hellipif I make no use of its rightness and consider only its rectilinear character, the proposition will apply equally to all angles with rectilinear sides. (A Commentary on the First Book of Euclid&rsquos Elements, Morrow 1970: 207))

The place of diagrams in geometry remained an issue into the early modern period. Major philosophical figures in the 17 th and 18 th centuries advanced positions on it. Anticipating the predominate modern view, Leibniz asserts:

&hellipit is not the figures which furnish the proof with geometers, though the style of the exposition may make you think so. The force of the demonstration is independent of the figure drawn, which is drawn only to facilitate the knowledge of our meaning, and to fix the attention it is the universal propositions, i.e., the definitions, axioms, and theorems already demonstrated, which make the reasoning, and which would sustain it though the figure were not there. (1704 New Essays: 403)

In the introduction to his Principles of Human Knowledge (1710, section 16), Berkeley reiterates 13 centuries later Proclus&rsquos take on the generality problem. Though one always has a particular triangle &lsquoin view&rsquo when working through a demonstration about triangles, there is &lsquonot the least mention&rsquo of the particular details of the particular triangle in the demonstration. The demonstration thus proves, according to Berkeley, a general proposition about triangles.

The most developed, and predictably most complex and difficult, account of geometric diagrams in the modern period can be found in Kant. Kant saw something of deep epistemological significance in the geometer&rsquos use of a particular diagram to reason about a geometric concept. In reasoning in this way, the geometer

considers the concept in concreto, although non-empirically, but rather solely as one it has exhibited a priori, i.e., constructed, and in which that which follows from the general conditions of the construction must also generally of the object of the constructed concept. (1781, Critique of Pure Reason, A716/B744.)

For contrasting views of what passages like these reveal about where diagrams fit in Kant&rsquos philosophy of geometry, see Shabel 2003 and Friedman 2012.

In the 19 th century geometry and mathematics as a whole underwent a revolution. Concepts far more abstract and general than those found in the Elements (e.g., non-Euclidean geometries, sets) emerged. Not only did questions about the nature of Euclid&rsquos diagrammatic method lose their urgency, the method came to be understood as mathematically flawed. The latter view found its most precise expression in the groundbreaking work of Moritz Pasch, who provided the first modern axiomatization of elementary geometry in Pasch (1882). In it, Pasch showed how the subject could be developed without reference to diagrams or even to the geometric concepts diagrams instantiate. The methodological norm guiding the work is nicely expressed in the following often-quoted passage:

In fact, if geometry is genuinely deductive, the process of deducing must be in all respects independent of the sense of the geometrical concepts, just as it must be independent of figures only the relations set out between the geometrical concepts used in the propositions (respectively definitions) concerned ought to be taken into account. (Pasch 1882: 98 emphasis in original. The translation here is from Schlimm 2010)

The norm has since entrenched itself both in mathematics and in philosophical discussions of mathematics. It is its entrenchment in the latter that Manders opposes in Manders 2008 [1995]. In the account he develops of ancient geometry, the necessity of consulting a diagram in a proof does not indicate a deductive gap. Rather, diagram and text together form a rigorous and deductive mathematical proof.

4.2 Manders&rsquo exact/co-exact distinction and the generality problem

4.2.1 The exact/co-exact distinction

To explain the division of labor between text and diagram in ancient geometry, Manders distinguishes between the exact and co-exact properties of geometric diagrams in Manders 2008 [1995]. Underlying the distinction is a notion of variation. The co-exact conditions realized by a diagram &lsquoare those conditions which are unaffected by some range of every continuous variation of a specified diagram.&rsquo Exact conditions, in contrast, are affected once the diagram is subject to the smallest variation. Roughly, a diagram&rsquos co-exact properties comprise the ways its parts define a finite set of planar regions, and the containment relations between these regions. A prominent exact relation is the equality of two magnitudes within a diagram. For example, only the slightest change to the position of CF in the diagram for proposition 16 is required to make the angles BAE and ECF unequal.

Manders&rsquo key observation is that Euclid&rsquos diagrams contribute to proofs only through their co-exact properties. Euclid never infers an exact property from a diagram unless it follows directly from a co-exact property. Relations between magnitudes which are not exhibited as a containment are either assumed from the outset or are proved via a chain of inferences in the text. This can easily be confirmed with the proof of proposition 16. The one inference that relies on the diagram is the second to last inference of the proof. The inference, specifically, is that angle ACD is greater than angle ACF. This, crucially, is based on seeing from the diagram that angle ACD contains angle ACF. There are many other relations asserted to hold in the proof. Though the diagram instantiates them, they are explicitly justified in the text. And with these relations, the relata are spatially separated magnitudes.

It is not difficult to hypothesize why Euclid would have restricted himself in such a way. It is only in their capacity to represent co-exact properties and relations that diagrams seem capable of functioning effectively as symbols of proof. The exact properties of diagrams are too refined to be easily reproducible and to support determinate judgments. As Manders puts it

The practice has resources to limit the risk of disagreement on (explicit) co-exact attributions from a diagram but it lacks such resources for exact attributions, and therefore could not allow them without dissolving into a disarray of irresolvably conflicting judgements. (Manders 2008 [1995]: 91&ndash92)

Manders&rsquo insights lead naturally to the idea that Euclid&rsquos arguments could be formalized in a manner similar to the way Venn diagrams have been formalized in Shin 1994. The co-exact information carried by Euclid&rsquos diagrams is discrete. When a diagram is consulted for this information, what matters about it is how its lines and circles partition a bounded planar region into a finite set of sub-regions. This opens the door to conceptualizing Euclid&rsquos diagrams as part of the syntax of Euclid&rsquos proof method.

4.2.2 The generality problem with Euclid&rsquos constructions

Realizing this conception in a formal system of proof amounts, as in Shin 1994, to specifying the syntax and semantics of diagrams. On the syntactic side, this means defining Euclid&rsquos diagrams as formal objects precisely, and giving rules whereby diagrams as formal object figure in derivations of Euclid&rsquos propositions. On the semantic side, this means specifying how derivable expressions are to be interpreted geometrically, or in other words how exactly are they to be understood as representing Euclid&rsquos propositions.

The semantic situation with Euclid&rsquos diagrams is thus different from that with Venn&rsquos. Venn diagrams are used to prove logical results. The inferences made with them are topic neutral. Euclid&rsquos diagrams on the other hand are used to prove geometrical results. The inferences made with them are topic specific. In particular, though the objects of plane Euclidean geometry are abstract (e.g., geometric lines are breadthless), they are still spatial. Consequently, issues surrounding the spatiality of diagrams and representational scope do not arise with Euclid&rsquos diagrams as they do, for instance, with Euler diagrams. In the case of geometry, in fact, the spatiality of diagrams counts in their favor. Spatial constraints on what is possible with geometric configurations are also operative with spatial Euclidean diagrams.

Nevertheless, as recognized in the philosophical commentary on Euclid&rsquos geometry from antiquity onward, there are with Euclidean diagrams issues of representational scope to contend with. What is the justification for treating properties of a single geometric diagram as representative of all the configurations in the range of a proof? How can a single diagram prove a general result? Manders&rsquo exact/co-exact distinction provides the basis for a partial answer. The co-exact properties of a diagram can be shared by all geometric configurations in the range of a proof, and so in such cases one is justified in reading off co-exact properties from the diagram. In a proof about triangles for instance, variation among the configurations in the range of the proof is variation of exact properties&mdashe.g., the measure of the triangles&rsquo angles, the ratios between their sides. They all share the same co-exact properties&mdashi.e., they all consists of three bounded linear regions which together define an area.

This is not a full answer because Euclid&rsquos proofs typically involve constructions on an initial configuration type. With the proof of proposition 16, for example, a construction on a triangle with one side extended is specified. In such cases, a diagram may adequately represent the co-exact properties of an initial configuration. But the result of applying a proof&rsquos construction to the diagram cannot be assumed to represent the co-exact properties of all configurations resulting from the construction. One does not need to consider complex geometric situations to see this. Suppose for instance the initial configuration type of a proof is triangle. Then the diagram

serves to represent the co-exact properties of this type. Suppose further that the first step of a proof&rsquos construction is to drop the perpendicular from a vertex of the triangle to the line containing the side opposite the vertex. Then the result of carrying this step out on the diagram

ceases to be representative. That the perpendicular falls within the triangle in the diagram is a co-exact feature of it. But there are triangles with exact properties different from the initial diagram where applying the construction step results in a perpendicular lying outside the triangle. For example, with the triangle

the result of applying the construction step is

4.3 The formal systems FG and Eu

And so, carrying out a Euclidean construction on a representative diagram can result in an unrepresentative diagram. A central task of formalizing Euclid&rsquos diagrammatic proofs is accounting for this&mdashi.e., providing with its rules a method for distinguishing general co-exact features from non-general ones in diagrammatic representations of constructions. The systems FG and Eu take two different approaches to this task.

Employing the method of FG, one must produce with a diagram every case that could result from the construction. A general co-exact relation of the construction is then one that appears in every case. FG&rsquos demand that every case be produced would, of course, be of a little interest if it did not also provide a method for producing them all. The method FG provides depends on the fact that lines and circles in the system&rsquos diagrams are defined in purely topological terms. Their resulting flexibility makes it possible to formulate and implement in a computer program a general method for generating cases. [9]

The lines and circles of Eu diagrams are not similarly flexible. Accordingly, it cannot resolve the generality problem via case-analysis as FG does. The central idea of its approach is to allow diagrams to hold partial information from the outset. Within an Eu derivation, the diagram produced by a proof&rsquos construction has an initial content consisting in all the qualitative relations of the proof&rsquos initial diagram. The qualitative relations concerning objects added by the construction cannot be read off the diagram immediately. Those that can be read off the diagram must be derived by the system&rsquos rules. [10]

The differences between the FG and Eu approaches to formalizing Euclid&rsquos constructions can be understood as representing different general conceptions of the role of diagrams in mathematics. FG embodies a conception where diagrams concretely realize a range of mathematical possibilities. They support mathematical inference by furnishing direct access to these possibilities. Eu in contrast embodies a conception where diagrams serve to represent in a single symbol the various components of a complex mathematical situation. They support mathematical inference by allowing the mathematical reasoner to consider all these components in one place, and to focus on those components relevant to a proof.


Orthogonality

Stephen Andrilli , David Hecker , in Elementary Linear Algebra (Fifth Edition) , 2016

Orthogonal and Orthonormal Bases

Theorem 6.1 assures us that any orthogonal set of nonzero vectors in R n is linearly independent, so any such set forms a basis for some subspace of R n .

A basis B for a subspace W of R n is an orthogonal basis for W if and only if B is an orthogonal set. Similarly, a basis B for W is an orthonormal basis for W if and only if B is an orthonormal set.

The following corollary follows immediately from Theorem 6.1 :

Corollary 6.2

If B is an orthogonal set of n nonzero vectors in R n , then B is an orthogonal basis for R n . Similarly, if B is an orthonormal set of n vectors in R n , then B is an orthonormal basis for R n .

Consider the following subset of R 3 : <[1,0,−1],[−1,4,−1],[2,1,2]>. Because every pair of distinct vectors in this set is orthogonal (verify!), this is an orthogonal set. By Corollary 6.2 , this is also an orthogonal basis for R 3 . Normalizing each vector, we obtain the following orthonormal basis for R 3 :

One of the advantages of using an orthogonal or orthonormal basis is that it is easy to coordinatize vectors with respect to that basis.

If B = (v1,v2,…,vk) is a nonempty ordered orthogonal basis for a subspace W of R n , and if v is any vector in W , then

Consider the ordered orthogonal basis B = (v1,v2,v3) for R 3 from Example 2 , where v1 = [1,0,−1],v2 = [−1,4,−1], and v3 = [2,1,2]. Let v = [−1,5,3]. We will use Theorem 6.3 to find [v]B.

Now, vv1 = −4,vv2 = 18, and vv3 = 9. Also, v1v1 = 2,v2v2 = 18, and v3v3 = 9. Hence,


Finite Dimensional Vector Spaces

Stephen Andrilli , David Hecker , in Elementary Linear Algebra (Fourth Edition) , 2010

Summary of Results

This section includes several different, but equivalent, descriptions of linearly independent and linearly dependent sets of vectors. Several additional characterizations are described in the exercises. The most important results from both the section and the exercises are summarized in Table 4.1 .

Table 4.1 . Equivalent conditions for a subset S of a vector space to be linearly independent or linearly dependent

Linear Independence of SLinear Dependence of SSource
If S = <v1,…, vn> and a1v1 + … + anvn = 0, then a1 = a2 = … = an = 0. (The zero vector requires zero coefficients.)If S = <v1,…, vn>, then a1v1 + … + anvn = 0 for some scalars a1, a2,…, an, with some ai ≠ 0. (The zero vector does not require all coefficients to be zero.)Definition
No vector in S is a finite linear combination of other vectors in S.Some vector in S is a finite linear combination of other vectors in S. Theorem 4.8 and Remarks after Example 14
For every vS, we have v ∉ span(S −<v>).There is a vS such that v ∈ span(S − <v>).Alternate characterization
For every vS, span(S − <v>) does not contain all the vectors of span(S).There is some vS such that span(S − <v>) = span(S). Exercise 12
If S = <v1,…, vn>, then for each k vk ∉ span(<v1,…, vk − 1>). (Each vk is not a linear combination of the previous vectors in S.)If S = <v1,…, vn>, some vk can be expressed as vk = a1v1 + … + ak − 1vk − 1. (Some vk is a linear combination of the previous vectors in S.) Exercise 22
Every vector in span(S) can be uniquely expressed as a linear combination of the vectors in S.Some vector in span(S) can be expressed in more than one way as a linear combination of the vectors in S. Theorem 4.9 and Theorem 4.10
Every finite subset of S is linearly independent.Some finite subset of S is linearly dependent.Definition when S is infinite

New Vocabulary

linearly dependent (set of vectors)

linearly independent (set of vectors)

Highlights

A set of vectors is linearly dependent if there is a nontrivial linear combination of the vectors that equals 0.

A set of vectors is linearly independent if the only linear combination of the vectors that equals 0 is the trivial linear combination (i.e., all coefficients = 0).

A single element set <v> is linearly independent if and only if v0.

A two-element set <v1, v2> is linearly independent if and only if neither vector is a scalar multiple of the other.

The vectors <e1,…, en> are linearly independent in ℝ n , and the vectors <1,x,x2,…, xn> are linearly independent in P n .

Any set containing the zero vector is linearly dependent.

The Independence Test Method determines whether a finite set is linearly independent by calculating the reduced row echelon form of the matrix whose columns are the given vectors.

If a subset of ℝ n contains more than n vectors, then the subset is linearly dependent.

A set of vectors is linearly dependent if some vector can be expressed as a linear combination of the others (i.e., is in the span of the other vectors). (Such a vector is said to be redundant.)

A set of vectors is linearly independent if no vector can be expressed as a linear combination of the others (i.e., is in the span of the other vectors).

A set of vectors is linearly independent if no vector can be expressed as a linear combination of those listed before it in the set.

A set of fundamental eigenvectors produced by the Diagonalization Method is linearly independent (this will be justified in Section 5.6 ).

An infinite set of vectors is linearly dependent if some finite subset is linearly dependent.

An infinite set of vectors is linearly independent if every finite subset is linearly independent.

A set S of vectors is linearly independent if and only if every vector in span(S) is produced by a unique linear combination of the vectors in S.


3. Alphabetic numerical notation

As soon as the order of letters in an alphabet became fixed, this opened a way to use the letters as numeric signs. Notwithstanding the fact that alphabets first emerged among Semitic peoples, first of all Phoenicians, it seems quite reasonable to suggest the Greeks being the inventors of alphabetic numeric notation ca. the sixth century BC (Chrisomalis 2010: 134). This system known as the Ionic or Milesian notation followed the Attic or Herodianic notation described in the previous section. The Greeks used three archaic letters to supplement their 24-letter alphabet expanding its capacity to denote unities, tens, and hundreds fully. Alphabetic numeric notation was borrowed by the Copts together with the alphabet, and also later by some other people, namely, Slavs, Armenians, Goths, etc.

In ancient Greek manuscripts the letters denoting numerals were distinguished from that for words by an overline, while modern practice is to add an acute-like sign on top-right of the numeral letter sequence. The archaic letters supplementing the Greek alphabet are: stigma (ϛ) or digamma (ϝ) for ‘6’, koppa (ϟ or ϙ) for ‘90’, and sampi (ϡ) for ‘900’, see Table 2.1. Thousands are marked by symbols for unities 1–9 preceded by a small stroke placed mostly to the bottom-left. Myriads (10,000s) are marked in a number of ways, in particular by the capital ‘M’ with the number of myriads placed above or before it (see Tables 2.1, 2.2). Another approach was to place a trema (dieresis or umlaut) above the letter for unities (Heath 2003: 18), e.g.

The numerals were written left to right, with letters for lower digits following that for higher ones, thus σμπ for 248 = 200 + 40 + 8.

Special symbols were used for fractions 1/2, 2/3, and 3/4 (see Table 2.5), while other unit fractions were written with a stroke on the top-left of the symbol corresponding to the denominator. This approach is however not a unique one and other ways to write fractions are also known (Heath 2003: 20–24).

Table 2.1: Alphabetic numerical notation

Note: Arabic alphabetic numerals are given for the western (Maghreb) system, with the eastern one in parentheses.

Table 2.2: Notation of large numerals in some alphabetic systems.

Asterisks denote suggested notation not confirmed in the available sources

The letters of the Coptic alphabet, being a descendant of Greek, were used to write numerals in a similar fashion. Some deviation can be observed in notation of thousands and myriads (Mallon 1907: 76–77), see Table 2.2: symbols correspond to 100 × 1,000 = 100,000, while denote 1,000 × 1,000 = 1,000,000, with another order is marked by an additional overline: for 10,000,000. It should be noted that, as it was usually for alphabetic systems of numerical notation, large numerals did not have a firmly established representation standard, and for instance Chrisomalis (2010: 136–137 and 148) cites only numerals up to 9,000 written just in the Greek fashion. The Coptic alphabetic numerals were mostly used from the fourth to the tenth centuries AD, but they still remain as a notation system within the Coptic Church.

Ethiopic number shapes are also of Greek origin, see Table 2.1. In this system, however, all the signs after ρ (100) are abandoned and larger numbers are formed by a multiplicative approach, with the number of hundreds placed before the sign ፻ for ‘100’. The next new sign was ፼, a ligature of two ፻, denoting 100 × 100 = 10,000. Occasionally, the symbol ሺ shi is used for ‘thousand’, but rather with western numerals, not with the Ethiopic ones. Large numerals can be obtained by the multiplicative principle (http://www.geez.org/Numerals/), see Table 2.2.

Two types of numerological notation are also associated with the Ethiopic script (http://www.geez.org/Numerals/Numerology.html). One was almost directly copied from Hebrew Gematria, see Table 2.3. Note that values from 500 to 900 did not have a unique sign in the Hebrew system. The sign for 900 in the Ethiopic Gematria system is taken from the additional zemede series of signs for labiovelars, as does the sign for 1,000 (see Table 2.2). The symbol for 10,000 is alef-sadis /’ə/, while all the previously mentioned ones belong to geez series /-ä/.

The complete table of the Ethiopic script is used in the Debtera (Halehame) system, see Table 2.4. The first series ending in /-ä/ correspond to numerals 1 to 800, while in subsequent columns these values are multiplied by 2, 3, etc. to 7 for the /-o/ series. Thus the maximum value in this system is 5600. However, it seems that only the numbers from the first series were used in numerological calculations ( http://www.geez.org/Numerals/Numerology.html ).

Table 2.3: Ethiopic Gematria compared to Hebrew and Greek alphabetic systems

For detailed description of Hebrew numerals, see Gandz (1932/33).

Table 2.4: Ethiopic Debtera (Halehame) system

In the tenth century AD an alphabetic system based on the cursive script evolved in Egypt. This system is known as “numerals of the Epakt”, Zimām numerals, or Coptic numerals (Chrisomalis 2010: 149–150). Except for the symbol shapes, the system is similar to Coptic and Greek alphabetic notations described above (see Tables 2.1, 2.2). The numbers were written left to right, with symbols for highest values place on the left. From available descriptions (Sesiano 1989 Goldstein & Pingree 1981) it does not become sufficiently clear how the numerals from ten million were formed. Zimām numerals survived as long as the seventeenth century, being later replaced by the Arabic positional numerals (Chrisomalis 2010: 152).

Arabic alphabetic notation spread in Africa together with Islam in the seventh century (Cajori 1928 Chrisomalis 2010). In main features, it resembles the Greek principle as the arrangement of letters mostly corresponds to the Greek order and not to that of the Arabic alphabet. The presence of the 28 th letter allowed the extension of the notation to thousands in a natural way, see Table 2.1. A simple multiplicative principle is known for the numbers over 1,000 (Cajori 1928: 29). Note that the Arabic alphabetic numerical notation is written the same direction as the script does, i.e. right to left with the highest values placed rightmost.

Rumi numerals known also as Fez numerals (from the city of Fez or Fes, Arabic فاس‎, a city in Morocco) were used around this area starting from the sixteenth century AD (Chrisomalis 2010: 171). This notation system originated in Spain among the Arabic Christians of Toledo in 12–13 th centuries. The numerals are written right to left. Due to cursive writing, the shapes of symbols vary a lot, some generic ones are shown in Table 2.1. Both shapes and structure of the Rumi numeral notation demonstrate influence from Greek, Coptic, Zimām, and Arabic alphabetic system. For instance, there is no mark for multiplication by 10,000, which is the case in Greek, Coptic, and Zimām systems, but not in the Arabic one, where only multiplication by 1,000 is relevant.

In the Rumi notation special marks were also used for fractions, see Table 2.5 (Lazrek 2006).

Table 2.5: Fractions in Greek and Rumi alphabetic numerical notation systems

There is a modern alphabet for Wolof with letters having numerical values. This script is briefly described in the following section. It is not known how the alphabetic numerals are used in this writing system and what are the principles of representing numbers over 100. For the symbol set with phonetic values of letters see two last columns in Table 2.1.


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