Single-Valued Functions vs. Many Valued Operations

Shraga YeshurunBar-Ilan University, Rajnmat-Gan Israel.

1. Till about the twenties of this century the teaching of thesecondary-school mathematics�at least in Europe�included twodistinct parts [5]:* algebra on one hand, geometry & trigonometryon the other hand. Both of them included a body of theorems, ap-plications, examples and exercises without paying much attention toany unifying principle. At that time a reform movement arose whichmade the concept of the function the unifying idea of school mathe-matics [4].At those times, as even today in some text-books (e.g. [6]) the

function was seen as related variation. The variable as its literalmeaning, was held to be capable of being varied. The independentvariable takes different values, one after the other, and to each valueof this independent variable belongs one value or belong more valuesof the dependent variable, i.e., the function may be single-valued ormultiple-valued. This viewpoint makes the graphical representationof the function in a Cartesian coordinate-system, very plausible.According to the graphical representation of the function (e.g., acircle, or the arc-sin-graph) it is possible of course to separate a curverepresenting a multiple-valued function into two or more branches,each corresponding to a single-valued function [2].

In all cases�we must emphasise it�the single-valuedness is only amatter of convention, and not a mathematical necessity. We couldspeak about multi-valued functions if we want to. In fact, in analyticgeometry the general practice is to write the "equation’7 in which toone value of x may belong two ^-values, i.e., the "equation57 is abivalued function.

In the last decade or so arose another search for elementary no-tions underlying most of the subjects of school mathematics, andthey were found in the theory of sets, groups, rings and fields [I],[3]. According to the terminology of the theory of sets only a single-valued function is named now "function77 whereas a multi-valuedfunction is included in the concept of "relation.77As a conclusion, today one looks at \/x as +V^ o^y? e.g., -\/4==

+2 only and \/4^�2. A further conclusion is that Vx being seenas a function, i.e., a single-valued function, its meaning as an inversealgebraic operation has been abandoned. So, for the sake of reachingunifying concepts in the most parts of school-mathematics, theexisting uniformity of the fundamental algebraic operations has been

* Numerals in brackets refer to the references at the end of this article.

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destroyed. This is regretable because many rules of the algebraicoperations (part of them definitions or postulates, part of themtheorems) and even further subjects in algebra are taught with lesseffort and with more efficiency if we don^t cease to see the root andespecially the square-root as one of the two inverse operations of thepower. In this case it must be \/4== – 2, because the inverse of either(-|-2)2=+4 or (-2)2==+4 should be V4. In short, we must distin-guish between the (single-valued) function y==\/’x and the inversealgebraic operation \/x, which is bivalued.

It does not seem possible to invent two completely different sym-bols for the two purposes, but I am not sure if it were not a betterpractice to use for^the single-valued function the symbol y=+\/xrather than y==\/x. If we do not want to use y=-\-^/x for writingthe single-valued function, we must rest on the belief that from thecontext it will always be clear whether -\/x is a single-valued func-tion or a bivalued inverse operation. If the teacher thinks that beliefnot to be a good practice, it remains for him to use – -\/^ for theinverse algebraic operation.

2. As many other authors, Adier points out (1): ^Mathematicswas taught in the past as a jumble of disconnected facts. But if weintroduce . . . ideas of mathematical structure, all the formerlydisconnected facts will fall into place as part of a coherent whole.They will begin to make sense and mathematics will be an easiersubject to learn^.This is true not only if we think the different parts of mathematics

as seen from a new, common and unifying viewpoint, but also if wesum up a series of rules (axioms and theorems) into one which in-cludes all of them. This will be a didactic gain which will expressitself both in lightening the burden on the memory and in makingeasier and safer the mistakeless application in any concrete case.In what follows we shall show some such ideas, not generally in

use in today’s mathematics teaching, which serve the above men-tioned purpose: summing up into one general rule several discon-nected facts. The common feature in all of them being: \/x must beunderstood in them as a bivalued inverse algebraic operation, ratherthan a single-valued function.

For easy reference we will order the seven algebriac operationsinto a 2X3 table; addition, multiplication and raising to power beingdirect operations, while subtraction, division, finding root and loga-rithmation are inverse operations. This observation gives the twocolumns of the table. On the other hand addition and its inverse,subtraction are the fundamental operations which cannot be derivedfrom other operations, and so they are named: operations of 1storder. Multiplication being repeated addition, and its inverse -divi-

Single-Valued Functions vs. Many Valued Operations 719

sion- being repeated subtraction, of equal terms they are named:operations of 2nd order. Raising to power, being repeated multiplica-tion of equal factors, it and its two inverse operations, are named:operations of 3rd order. The hierarchy of the operations is summarizedin Table 1.

TABLE 1. HIERARCHY OF ALGEBRAIC OPERATIONS

Order Direct Operations Inverse Operations

1st Addition Subtraction2nd Multiplication Division3rd Raising to power Finding root & logarithmation

By the way, from Table 1 it becomes clear why there is no 4th-orderalgebraic operation: only an associative operation can be repeatedand so, if an nth order direct operation is associative, there exists an(w+l)st order operation. The 1st and 2nd order direct operationsare associative but the 3rd order one is not, and so there exist 2ndand 3rd order operations, but no 4th order one. The table gives someinformation about commutativeness, too. The 1st and 2nd orderdirect operations are commutative, therefore they have only oneinverse operation. The 3rd order direct operation is not commutativeand so it has two distinct inverse operations.

2.1. Still important and interesting is the case of the distributivelaw. We can observe that an operation is distributive over anotheroperation if, and only if, the latter is exactly one order lower than theformer.We need not deal in this context with the fact that part of this

observation is definition, and another part is theorem which can beproved. Our concern in this context is that this single rule comprisesa lot of rules (definitions and theorems):

(a + b)c == ac + be

a + b a b

c c c

(ab)71 == a^

\^ab = v^’ ^/bBut

(a + bY ^ c^ + ^because addition is two orders lower than raising to power, and notexactly one. Also, ab/c^a/c-b/c because multiplication is of thesame order as division and not exactly one order lower, etc.

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If we teach according to this one rule�one can avoid a lot of fre-quently repeated mistakes.

2.2. Another use of the table of the hierarchy (Table 1) is thesummarizing up of the rules of the exponents into one single rule:"The operation in the exponent is lower by exactly one order". Thissingle rule comprises the following:

ak� = ^-1a1

(^)i = ^i

| v^-| == | ^/iIn the case of the latter the absolute-value-sign comes because of thepossible double valuedness of y^- As a contrary, akll is alwayssingle-valued and it exists only for a>0. On the other hand, this ruleis true either if k/1 is an integer and then it is a theorem, or if k/l is afraction, then it is a definition. But a^’a1^^1 because the operationin the exponent must be lower by exactly one order and so, insteadof multiplication of the whole algebraic expression, we use addition inthe exponent. And again, by teaching of this single rule we can avoida lot of frequently repeated mistakes.

Furthermore, since the logarithm, too, is an exponent, this rule isvalid for logarithms too: "The operation with the logarithm, beingan exponent, is lower by exactly one order"�and so:

log (a + b)

log (a � b)

are not accomplishable because there exists no operation whose orderis lower than 1.

log ab = log a + log b

log {a/b) �== log a � log b

log (a^ == k � log a

log ^ = (log a)/k {a, b > 0)

As up until now, again, by teaching a single rule we can avoid a lotof frequently repeated mistakes.

2.3. A curious use of Table 1 is the distinction between arithmeticand geometric mean: if we perform firstly a 1st order direct operation,and afterwords a 2nd order inverse operation, -this is the easiestpossible combination of a direct and an inverse operation of differ-ent orders. The result is named "Arithmetic" Mean. If we perform

Single-Valued Functions vs. Many Valued Operations 721

firstly a 2nd order direct operation, and afterwards a 3rd order in-verse operation, we arrive at the second possible combination of adirect and an inverse operation of different orders, the result of whichis named ^Geometric" Mean.

2.4. By a further exploitation of Table 1 we can put another groupof disconnected facts into place as part of a coherent whole. For thisaim we mention again that the five first operations (i.e., the fouroperations of 1st and 2nd order and the direct operation of 3rdorder) are single-valued. Finding a root may be multi-valued evenin the real domain, while the last operation, the logarithmation, issingle-valued if only the real domain is considered i.e. if the resulttoo, must be real.Now there is no question that in solving equations, or more

generally, in treating equalities, we can perform any single-valuedoperation on both sides of the equality, but we can not perform amulti-valued operation and so we can comprise all the distinct rulesusing the notion of multi-valued operation.More complicated is the conglomerate of the rules governing the

treatment of inequalities. Usually they are taught as a list of rules ofpermitted operations in the treatment of inequalities, and perhapsanother list of the forbidden operations (e.g., [7]). Instead of sucha list we can teach a coherent whole if we observe which of the singlevalued operations are monotone increasing in the real domain, whichare monotone decreasing and which are not monotone at all. Weconfine ourselves to the real domain only, because inequalities(greater than or less than) do not exist in the complex domain. Fur-thermore, it is possible in some cases (e.g., in the case of raising to aneven power) to divide the real domain so that over one part, theoperation is monotone decreasing, over the other one, it is monotoneincreasing, although over the whole real domain it is not monotone atall.

According to this observation we can resume all the rules governingthe treatment of equalities, into one comprehensive rule as follows:Any single-valued operation, and only such an operation is per-

mitted on both sides of an equality.For the treatment of an inequality the single-valuedness of the

operation is a necessary but not sufficient condition. Any single-valued and monotone operation, and only such an operation is per-mitted on both sides of an inequality. Furthermore, if the operationis monotone increasing (e.g., multiplication by a positive number, orraising to an even power if it is known that both sides of the in-equality are non-negative, etc.), the direction of the inequality(>or<) remains unchanged; if the operation is monotone decreasing(e.g., multiplication by a negative number or raising to an even

722 School Science and Mathematics

power if it is known that both sides of the inequality are non-positiveetc.), the direction of the inequality must be changed (>to< and<to>).

2.5. Appendix. It is possible to exploit our Table 1 for the exten-sion of the number notion.

If we want the inverse operations to make sense in any case, wecan define zero and negative numbers by subtraction, fractions bydivision, irrational and imaginary numbers by finding square-root.This is a well-known use of Table 1, but it is not connected to theproblem of the multi-valued operation, and so we shall not insist onit further.

REFERENCES[1] ADLER, IRVING, The Cambridge Report, The Mathematics Teacher LIX,

214, 1966.[2] COURANT R., Differential and Integral Calculus, 2nd ed, Blackie & Son,

1963, pp 16ff.[3] DAVIS and Al, Goals for School Mathematics (Cambridge Report) 1963.[4] KLEIN, FELIX, Elementary Mathematics From an Advanced Standpoint,

Dover, w/o year. All the book and especially the Introduction of the volumeGeometry.

[5] New Thinking in School Mathematics, OEEC, 1961.[6] SPIEGEL, MURRAY R., Statistics, Schaum, 1961, p. 1.[7] SPIEGEL, MURRAY R., College Algebra, Schaum, 1956 p 167-8.

1,000 NEW LEPROSY CASES IN MEXICO LAST YEARThe Mexican Department of Health reports 1,000 new cases of leprosy in the

republic last year. These cases brought the registered total to more than 18,000persons.The actual leprosy count may be more than double this figure, according to the

Department of Health, since in hinterlands citizens avoid coming to clinics fortesting and treatment.

In Indian communities attitude to mobile brigades patroling endemic zones isdownright hostile. A total of 22 special mobile brigades cover the "leprosy zone"which includes San Luis Potosi, Jalisco and the state of Oaxaca, the Departmentof Health report said.

LACK OF LEGAL STATUS HOLDS BACK SEA MINERSMineral deposits in the ocean floor could be mined today if the mining com-

panies could be assured of their legal status there, according to the director of theU. S. Bureau of Mines. ^There are so many uncertainties in the ocean," notes Dr.Walter R. Hibbard Jr., that the mining industry has been very hesitant to followthe petroleum industry out into the sea, even near shore.

Dr. Hibbard discussed the legal aspects of sea-floor mining at the SecondAnnual Summer Conference on the Future of the Sea’s Resources at the Univer-sity of Rhode Island.

(<! think the Bureau of Mines’ role is to set up a demonstration operation,"Hibbard feels. "This is what we hope to do in the Bering Sea off Nome, Alaska."