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2 TEXT 1 Machine Translation Scientists say that our human language is a sign system, the elements of which may be used to express the content of any other system signs. For example, the signs of traffic signalization can always be translated into ordinary language. A most simple system is the multiplication table: 2 x 2 = 4; 2 x 3 = 6; etc. In French, Russian, Iakutian and other languages this simple truth will sound differently, while the language of numbers is comprehensible to all. By replacing words with members, linguistic signs by the signs of another system, we have at our disposal a universal language, the language of mathematics. Numbers express only one concept. Compare the number 1 and the word “one”. The number has a single meaning, the word has several meanings. Thanks to the system of numbers and mathematical symbols we save a lot in space and time. Logicians, physicists, chemists set up special system of signs used to express the complex concepts, ideas and facts of science Engineers use engineering drawings. Geographers use the language of maps. Compare the convenience and exactitude of the map with any verbal description of location and layouts. Compare the drawings of housing construction with a verbal description of how to build a structure. Cybernetics has brought to man a reliable and true helper in the form of computers. Everything expressible in one language can be translated into another However, there are many things that do not submit to the exact language of science. That’s why translating human knowledge into a machine language is so far only possible in the field of the exact sciences such as mathematics, logic, physics, chemistry and engineering. Machines are not yet capable of understanding our human polysemous language. The idea of machine translation originated long before the birth of cybernatics and electronic calculating machines. But it was only the electronic computers that could put machine translation on a firm basic.
I. Pronounce the following words: human sign content multiplication exactitude verbal reliabler submit
['hjumwn] [sain] > NQWwQW@ [m∧ltipli'kei∫n] [ig'zæktitju:d] >Yw EwO@ [ri'laiwbl] [s∧b'mit]
replace have at one’s disposal save logician capable polysemous originate calculate
[ri'pleis] >GLV SRX]wO@ [seiv] [lo'd i∫LwQ@ >NHLSwEO@ >SROL VHPRXwV@ >wULGLQHLW@ ['kælkjuleit]
3 II. Answer the following questions:
1. 2. 3. 4. 5. 6. 7.
What is human language? Is the language of numbers comprehensible? What language do you have at our disposal? How do we save a lot in space and time? What has cybernetics brought to man? Are machines capable of understanding our human polysemous language? When did the idea of machine translation originate?
TEXT 2 Machine Translation II Today machine translation is a matter of prime economic importance. The problem is being attacked by logicians, engineers, linguists and mathematicians. The problem is by no means as easy one. Human language is much too polysemous and rich; it is difficult to adapt such an instrument to the rigid language of electronic computers. What is a translation? What can be translated and what cannot? All the letters of a language can be encoded. Any text can be translated into this code, but it will never be termed a translation. The fist attempts at machine translation were only experimental and did not represent serious works of translation. The texts were rather primitive and were based on a small vocabulary with highly limited rules of grammar. The problem of creating a machine language capable of taking in the entire wealth of the living language is very complicated. It must not be indifferent to the style of description, the emotional undertones and linguistic subtleties. Let’s take an example: It reads identically to such a synonymous spectrum as: “woman, female, date, petticoat, skirt.” All these words would be recorded by a single character of the meaning - code, which describes an individual of the Homo-sapience female sex, and that is all. Machine translation is being studied by researchers in the United States, Russia and other countries. They hope to overcome the numerous linguistic and stylistic difficulties and create a machine that will make it possible to translate from one natural language into another just as easily as it makes translation from the language of one science to that of another. The language of the machine is the language of numbers and exact facts. Try to translate “Hamlet” or modern poetry into machine language and you get nothing. Poetry, music and painting will always remain in the domain of man only.
4 I. Pronounce the following words: prime flexible rigid encode attempt identically
>SUZLP@ [fleksibl] ['rid id] [in'koud] [w tempt] [aidwntikwli]
wealth complicated undertone subtlety female domain overcome
[welθ ] [kwmpli'keitid] [∧ndw toun] ['s∧tlti] [fi'meil] [dw mein] [ouvwk∧m]
II. Answer the following questions: 1. Who is the problem of machine translation being attacked by? 2. Is it difficult to adapt human language to the rigid language of electronic computers? 3. Can any text be translated into this code? 4. Which problem is very complicated? 5. Where is machine translation being studied? 6. What do researchers hope? 7. What is the language of the machine?
TEXT 3 The Wheel of Aristotle Paradoxes have often been used to add interest to logic and proof. One which has often been used is known as the wheel of Aristotle. We are given two circular wheels A and B with different radii and with the smaller wheel B fixed to A so that the centers coincide. Then one wheel cannot rotate through any angle without the other rotating through the same angle. Pick a point P on the circumference of A and let the wheel roll in a straight line on a flat surface through 360° with the wheel starting with point P1 on the surface and ending with point P2 on the surface. Then the line segment traced on the plane equals the circumference of the circle A. But the wheel B has also completed exactly one revolution and so has traced out its circumference also. Does this mean that A and B have exactly the same circumference even though they were assumed to be different? When a student meets a paradox like this, his first attempts are to refute it. This can be done in two ways: (1) by finding a fallacy in the proof, or (2) by finding a fallacy in the construction. An example of the first type is the following proof
5 that 1 = 0 let x = 1 then x2 = x, and x2 – x = 0 Factoring, x (x – 1) = 0 Dividing, x = 0 : (x – 1) = 0, so x = 1 = 0 of course, the fallacy with this proof is division by x – 1 or division by zero which is impossible in our number system. Assume that we make a physical model of the wheel of Aristotle as we described it earlier. Now we wrap string about the wheel B equal to its circumference and about the wheel A equal to its circumference. Finally, we fix one end of each string to a vertical peg and the other and to the wheel, and let the wheels roll on a flat table. If the wheels roll with the string attached to B held tightly then we notice that the string attached to A becomes compressed and begins to pile up behind A. This kind of phenomenon was not taken into account in our proof! Indeed, the compressibility or stretchability of the string was not considered at all. The main difficulty with the wheel of Aristotle seems to be a misinterpretation of what was really proved. By ignoring the compressibility of our ideal mathematical lines, we were led to a contradiction. The proof gives an indication of the difficulty of representing problems of the physical world by mathematical methods. It shows that one must be extremely careful as to the assumptions made when a problem is posed. I. Pronounce the following words: ['pærwdoks] paradox ['pruf] proof [wi:l] wheel ['sw:kjulw] circular ['reidiai] radii [kouin'said] coincide ['æJO@ angle circumference [sw k∧mferwns] surface ['sw:fis] refute [ri'f ju:t] 1. 2. 3. 4. 5. 6. 7.
fallacy factoring flat phenomenon take into account compressibility stretchability consider contradiction be posed
['fælwsi] ['fæktwUL@ [flæt] [fi'nminwn] [w kaunt] [kwmpresi'biliti] [stret∫w biliti] [kwn'sidw] [kwn'trw'dikwn] [pouzd]
II. Answer the following questions: When have paradoxes been used? Which is well – known? What equals the circumference of the circle A? What is the first reaction of a student when he meets a paradox like this? How can he refute the paradox? What is the fallacy of this proof? Does it show that one must be careful as to the assumptions made when a problem is posed?
6 TEXT 4 Numbers While the beginning of our number system dates back to the remotest antiquity, where symbols, which we call positive integers, were used as an aid in counting, it was not until the nineteenth century that the system which we know today was completed. As an aid in tracing our own knowledge of this number system, let us make use of accompaning diagram.
integers rational read complex
fractions
irrational
imaginary
The first numbers we use are positive integers, and fundamental fact that there is the first integer unity, but not a last is soon established. Later positive fractions or numbers which can be expressed as the ratio of two of these integers, are used and understood. Then it is seen that these integers and fractions can be negative as well as positive. The division point between the positive and negative members, which is the position from which we start to count in either direction, is found to be occupied from all others in that we are not allowed to use it as a divisor. The positive and negative integers and fractions, together with zero, are called rational members. In contradiction to the rational numbers we find the irrational, which are defined as numbers that cannot be expressed as the quotient of two integers. The 2 , − 3 and π are examples of such numbers. The two classes of numbers, rational and irrational form the real number system, and are the ones which we shall use in the first part of the course. Later shall study such numbers as − 2 , - − 1 etc., which we call imaginaries and finally it will be seen that the basic system of all of the numbers which we shall use is the complex, in which the real and imaginaries are included as special cases. For example, 2 + − 3 is such a number and we see that is composed of a real and imaginary part.
7 Absolute Values The absolute, or numerical value of a real number is the value of the number without respect to sign. Thus the absolute value of a positive number is the number itself, whereas the absolute value of a negative number is the positive value of the number. The absolute value is denoted by two vertical bars inclosing the number. 0 = / 0 / −4 = / 4 / Terms The parts of an expression separated by the signs + and −, each part being taken with the sign immediately preceding it, are called terms. Thus the expression 2a – 3b + c is composed of three terms: 2a, −3b and c. Terms may be grouped by means of the parenthesis ( ), the brackets [ ], the braces { } and the vinculums. They show that the terms inclosed by them are to be treated as a single number. It a + sign appears before a symbol of grouping, it indicates that the sign of the terms inclosed by it remains the same. It such a symbol is preceded by a − sign it shows that the inclosed term is to be changed when the symbol is removed. I. Pronounce the following words: integer real imaginary unity ratio quotient vinculum
[intid w] [riwl] [im'æd inwri] ['juniti] ['rei∫iou] ['knouwnt] [vinkjulwm]
rational irrational fraction diagram absolute parenthesis bracket brace
['ræwnl] [ir'ræwnl] [frækQ@ [daiw græm] ['æbswlu:t] [pw ræwVLV@ ['brækit] [breis]
II. Answer the following questions: 1. What integers - positive or negative – were used at the very beginning of our number system? 2. What did negative integers mean? 3. What is the division point between the positive and negative numbers? 4. What is the difference between the number zero and other numbers? 5. What numbers are called rational? 6. What numbers do we call irrational? 7. What numbers form the real number system? 8. What is the absolute value of a real number?
8 TEXT 5 Involution and Evolution Involution, or raising of a given number to a power, is the repeated multiplication of that number by itself. Thus if “b” is multiplied by itself (n – 1) times, we have bn = b × b × b ×… n factors. Here “n” is called the exponent or index and is positive integer, “b” the base, and “b n” the n-th power of “b” Ex.
24 = 2 × 2 × 2 ×2 = 16 (−4)3 = (−4) × (−4) × (−4) = −64
Evolution is one of the inverse of involution. It is the operation of finding a root of a number. Thus if xn = b we find the value of “x” by extracting the n-th root of “b”. If x2 = 4, then x = ± 4 = ±2 . For, on substiluting these values of “x” in the original equation, we obtain (2)2 = 4, (−2)2 = 4 Polynomials. An algebraic expression such as 6x5 − 4x3 + 7x2 ± 8 is a polynomial or integral rational expression in the letter “x”. It is composed of one or more terms, each of which is either on integral power of “x” multiplied by a constant or a constant, which is free of “x”. The constant multipliers: 6, −4, 7 are called coefficients; the upper numbers: 5, 3, 2 exponents; 8, the constant term. The polynomial is of the fifth degree in “x” since 5 is the highest exponent appearing in the expression. An expression such as 2x3y + 5x4yz2 − 10xyz + 3x + 9 is a polynomial in “x”, “y” and “z”. The fifth term of this expression is of the third degree of “x”, of the first degree of “y”, and of the forth degree of “x” and “y”. The degree of a polynomial in several letters is the highest degree that any single term has in those letters. Thus the above expression is of the seventh degree in “x”, “y” and “z” since the sum of the exponents of the second term is seven. If a polynomial contains but one term, it is called a monomial; if two, a binomial; if three, a trinomial; etc.
9 The Fundamental Operations Applied to Polynomials 1) To add polynomials, place them in such a way that like terms fall under each other, and add the coefficients in each column to find the final coefficient of the term. Ex. Add 10x2y2 + 6xy − 3x2y − 2x; 3x2y2 + 2xy; 4xy − 2x2y + 3x + 4 10x2y2 + 6xy + 3x2y − 2x 3x2y2 + 2xy 4xy − 2x2y + 3x + 4 The sum is 13x2y2 + 12xy − 5 x2y + x + 4 2) To subtract one polynomial from another, place the terms of the subtrahend under the like terms of the minuend, change the signs of the terms of the subtrahend, and add. Ex.
Subtract x2+ + 2xy − 5x2z + 6x − 3x2
from
5xy + 2z + 2x2 + 5x − 3 2x2 + 5xy + 2z + 5x + 3 + 6x − 5x2z − 3z2 x2 + 2xy
The difference is x2 + 3xy + 2z − x + 3 + 5x2z + 3z2 3) To multiply one polynomial by another, multiply each term of one by every term of the other and add the products thus obtained. Ex. Multiply x2 − 2xy + y2 + 2 by 2x − 3y 2x − 3y 2x3 − 4x2y + 2xy2 + 4x − 3xy + 6xy2 − 3y2 − 6y The product is
2x2 − 7x2y + 8xy2 + 4x − 3y2 − 6y
10 4) To divide one polynomial by another, arrange both dividend and divisor in ascending or descending powers of some letter common to both and write the quotient as a fraction. If the degree of the denominator is greater than the degree of the numerator in the letter chosen, leave the quotient as a fraction. If the degree of the denominator is less than or equal to the degree of the numerator in the letter chosen, express the result as a quotient plus a fraction. The numerator of the fraction is called the remainder and is of lower degree in the letter chosen than the original divisor. This method of division is called long division. The rule may be stated thus: divide the leading term of the dividend by the leading term of the divisor, obtaining the first term of the quotient. Multiply each term of the divisor by this term of the quotient and subtract the product of the dividend. The remainder found by the subtraction is used as the dividend and the process is repeated. The work is continued until a remainder is reached which is of lower degree than the divisor in the letter of the arrangement. It is to be understood that the form of the quotient and remainder depends upon the choice of the letter of the arrangement. Ex. Divide x2 + 8xy + 4x2 by x + 2y a) Choose “x” as the letter of arrangement. We have x + 6y x + 2y x2 + 8xy + 4y2 x2 + 2xy 6xy + 4y2 6xy + 12y2 −8y2 Hence
x
2
+ 8xy + 4 y x + 2y
2
= x + 6y -
8y
4
x + 2y
where x + 6y is the quotient and −8y4 is the remainder. b) Choose “z” as the letter of the arrangement. We have
11 2y + 3x 4y2 + 8yx + x2 4y2 + 2yx
2 (y + x)
6yx + x2 6yx + 3x2 −2x2 Hence 4 y 2 + 8\[+[ + 2 \+[ [ 2y + x \+[ were 2y + 3x is the quotient and −2x2 is the remainder. In any case on division if the remainder is zero the division is said to be exact. I. Pronounce the following words: involution evolution exponent index binomial trinomial finite similarly
[invw lu:Q@ [ivw lu:Q@ [eks'pounwnt] ['indwks] [bai'noumjwl] [trai'noumjl] ['fainait] ['similwli]
II. Answer the following questions: 1. 2. 3. 4. 5.
What must you do to add polynomials? What do you do to subtract one polynomial from another? What is the way of multiplying one polynomial by another? What is the order of dividing one polynomial by another? What method is called the method of long division? TEXT 6 Equation
An equation is a statement expressing the equality of two expressions. If one of the expressions is exactly the same as the other or can be transformed into the other by means of the elementary operations of algebra, the expressions are said to be identically equal. In this case we have an identical equation or identity. If the two
12 expressions are finite and are identically equal, then they have equal values for all values of the unknown letters which occur in them. Thus, where the sign of identity is x2 + 2 (x + 3) − 5 = (x + 1)2 For
x2 + 2 (x + 3) − 5 = x2 + 2x + 6 − 5 = x2 + 2x + 1 = (x + 1)2
The identity is true for all values of “x”: Again (x + y)2 = x2 + 2xy + y2 and is true for all values of “x” and “y”. If two expressions form an equation and are not identically equal, they are said to form a conditional equation. Thus x − 5 = 15 − 3x is an equation of condition in one unknown letter. To solve the equation is to find the value or values of this letter which will make the two sides of the equation the same. Similarly the equation 2x + 3y = 5 is a conditional equation in two unknowns. An equation is one unknown has a finite number of solutions or values of the unknown, which will satisfy the equation. Thus x − 5 = 15 − 3x has the single solution 5 since this is the only value of “x” which will make the two sides of the equation equal. The highest exponent of “x” which occurs in this expression is 1: hence the equation is called first degree or simple. Again x2 − 1 = x + 1 is a second degree or quadratic equation in “x”, having −1 and 2 as solutions since these and no other values of “x” will satisfy the equation. It is not necessary that an equation has a solution, since the statement of condition may be such that no number can be found that will satisfy it. For instance, no value of “x” will satisfy x + 2 = 4 + x. I. Pronounce the following words: equation equality finite occur conditional quadratic
[ik'weiQ@ [ik'wo:liti] ['fainait] [w kw:] [kwn'diwnl] ['kwo:'drætik]
II. Answer the following questions: 1. 2. 3. 4.
What is an equation? When do the two expressions have equal values for all values? When do the two expressions form a conditional equation? Is it necessary that an equation has a solution?
13 TEXT 7 Variables and Constants A variable is a quantity or a symbol representing a quantity which is capable of change. It is usually represented by one of the last letters of the alphabet such as x, y, z. A constant is a number, or a symbol representing a number, which is incapable of change throughout a given investigation. Some constants, such as 2, −3, 5 , π etc., retain the same value under all circumstances. These are called absolute constants. On the other hand, the first letters of the alphabet, such as a, b, c, d, etc., are symbols used to represent constants which do not change in any given problem, but may vary in value from one problem to another. Such constants are called arbitrary constants. Ilustration. In the expression 3ax + 2y − 3z − 5b + c, we say that x, y and z are variables; 3, 2, 3 and 5 are absolute constants; and a, b, c are arbitrary constants. The symbols used to represent variables and arbitrary constants are not limited to the letters of the alphabet listed above, but may be the letters of any alphabet, the first letter of the name of the quantity under consideration, or any other symbol where the context of the problem makes the meaning clear. For example, g, t, a, v, T are symbols in common use representing gravity, time, acceleration, velocity and temperature respectively.
The Function Idea Experience teaches that most of the things around us are subjects to change and that a change in one usually causes a change in another. The seasons change according to the earth’s position with respect to the sun; atmospheric pressure decreases as we go above sea level; the price of an article of goods increases as the supply decreases, etc. In each of these illustrations we are aware of a basic relationship existing between the variables which may or may not lend itself to mathematical representation. This leads us to the definition of a function. Definition. When one variable is so related to another that for every value of the first there can be determinated one or more values of the second, the second is said to be a function of the first. The first variable of our definition is called the independent, and the second, the dependent. These names seen, to be reasonable, from the very nature of the definition. Thus water pressure is a function of the depth below the surface of the water, and it is clear that pressure depends upon the depth and is dependent variables. On the other hand, we can compute pressures at any depth we desire; hence, depth is thought of as the independent variable. It is to be understood that many functions involve more than one independent variable. For instance, the return of a unit of invested principal depends upon the rate
14 of interest paid as well as upon the time the principal is invested. In any case we have two independent variables. In general we say that the definition of a function may be extended to include any number of independent variables. Functions may be represented by means of equations, such as A = x2, y =
x , z = w2 + 5w − 4, 2
which show the functional relationship between the dependent variable on the left and the independent variable on the right on by a table of values, such as −20 −4
C E
−10 14
0 32
20 50
30 86
40 104
50 122
where F (degree of temperature, Farenheit) is shown as a function of C (degrees of temperature, Centigrade). In many cases the roles of the variables may be interchanged. For example, the second equation given above may be written x = 2y, and x now is expressed as a function of y. Again we may consider the values of the table as representing C as a function of F. We observe at once that this interchange of roles does not impair the definition of a function. In any particular investigation, the nature of the problem under consideration will determine which variable is to be taken as a function of the others. II.
Pronounce the following words:
variable constant quantity investigation acceleration velocity involve
[‘vεwribl] [‘konstwnt] [‘kwontiti] [investi’geiQ@ [‘ækswlw reiQ@ [vi’lositi] [in’volv]
throughout retain circumstances ordinarily arbitrary experience pressure
>UX¶DXW@ [ri’tein] [‘sw:kwmstwnsiz] [‘o:dinwrili] [‘a:bitrwri] [iks’piwriwns] [‘prew]
II. Answer the following questions: 1. 2. 3. 4. 5. 6.
What is a variable? What letters is a variable usually represented by? What is a constant? What constants are called absolute (arbitrary)? Can you give any examples of dependence one quantity upon another? What variables do you know?
15 TEXT 8 Low Level programming Languages: Machine and the Assembly Languages 1. In order to communicate with each other, men use languages. In the same way, “languages” of one sort or another are used in order to communicate instructions or commands to a computer. 2. When the user wishes to communicate with computer, he uses a spectrum of languages: English Fortran Algol Assembly language Mnemonic machine language Machine language 3. A machine language which is sometimes called as a basic programming language on autocode refers to instructions written in a machine code. This machine code can be immediately obeyed by a computer without translation. The machine code is the coding system adopted in the design of a computer to represent the instruction repertoire of the computer. The actual machine language is generated by software, not a programmer. The programmer writes in a programming language which is translated into the machine language. 4. A mnemonic machine language uses symbolic names for each part of instruction that is easier for the programmer to remember than the numeric code for the machine. A mnemonic is an alphanumeric name, usually beginning with a letter rather than a number to refer to fields, files, and subroutines in a program. “For example, the operation “multiplication might be represented as MULT, the “load” instruction as L, or DISP NAME ADDR in the mnemonic form means” display name and address”, etc. 5. An assembly language is the most machine dependent language used by programmers to-day. They are four advantages to using an assembly language rather than machine language. They are the following: 1) it is mnemonic 2) addresses are symbolic, not absolute as in a machine language 3) reading is easier 4) introduction of data to a program is easier. 6. A disadvange of assembly language is that it requires the use of an assembler to translate a source program into object code (program) in order to be directly understood by the computer. The program written in assembly language is called an assembler. The assembler usually uses such instructions as A (ADD), L (LOAD), St (STORE), start, test, begin, using, BALR (Branch and Link Register), DC (Define Constant), DS (Define Storage), END, etc. 7. The USING instruction is a pseudo-op. A pseudo-op is an assembly language instruction that specifies an operation of the assembler; it is distinguished from a machine-op which represents to the assembler a machine instruction. So, USING indicates to the assembler which general register to use as a base and what its
16 contents will be. This is necessary because no special registers are set aside for addressing, thus the programmer must inform the assembler the address contained to the base register. The assembler is thus able to produce the machine code with the correct base register and offset. 8. BALR is an instruction to the computer to load a register with the next address and branch to the address in the second field. It is important to see the distinction between the BALR which loads the base register, and the USING which informs the assembler what is in the base register. Hence, USING only provides information to the assembler but does not load the register. Therefore, if the register does not contain the address that the USING says it should contain, a program error may result. 9. START is a pseudo-op that tells the assembler where the beginning of the program is and allows the user to give a name to the program, e.g. it may be the name TEST. END is a pseudo-op that tells the assembler that the last card of the program has been reached. 10.Note that in the assembler instead of addresses in the operand fields of the instruction there are symbolic names. The main reason for assemblers coming into existence was to shift the burden of calculating specific addresses from the programmer to the computer. I. Pronounce the following words: mnemonic autocode obey adopt repertoire alphanumeric assembly assembler
[ni:'monik] ['o:tokoud] [w bei] [w dopt] ['repwtwa:] [ælfwnju:'merik] [w sembli] [w semblw]
source specify distinguish relative offset existence shift
[so:s] ['spesifai] [dis'tiηwi∫] ['relwtiv] ['ofset] [ig'zistwns] [LIW@
II. Answer the following questions: 1. 2. 3. 4. 5. 6. 7. 8.
What spectrum of languages does the user have at his disposal? Which languages do you call “low level”? When is the mnemonic form of the machine language used and why? Does the assembly language use symbolic addresses? What advantages to using the assembly language do you know? Which program can the computer directly understand? What is an assembler? What is a source program?
17 TEXT 9 The Hazards of Sets 1. The use of the set concept promises clarification, simplification, and unification in our teaching of mathematics. As teachers of mathematics, we have all tried to have our students appreciate the unifying principles that bind certain ideas into a mathematical whole. Where can we find a better unifier than the concept of set? 2. Can the term “set” be defined? This is a matter of choice. To introduce a discussion of set theory, one could choose to define “set” as follows: “y is a set < ∃ x) (x ∈ y Vy = 'V)”. But this is clearly unsuitable for the secondary school, where the aim is to teach set language and notation, but not set theory. At this level, it is common practice to take the term as undefined, just as “point”, “line”, “plane” are undefined terms in geometry. 3. Is every collection a set? Just as a dot is an intuitive picture of a point, so a collection of items is an intuitive picture of a set. However, a set is not that which we usually refer to in our everyday speech, when we speak of a large number of people, of ships, or of things. Groups of people with blue eyes, of big ships, and of things in that pile, do not quality as sets because of ambiguous cases that may arise. For example, does on individual with blue-green eyes belong to a group described as people with blue eyes? 4. In what way is the term “set” restricted? Although the term “set” may be taken as undefined, it must be restricted to avoid ambiguities. When we speak of “sets”, we should be referring only to those collections what consist of “definite distinct objects”. The inventor of set theory Georg Cantor put it this way: “A set is a bringing together into a whole of definite well-distinguished objects of our perception or thought”. Another way of saying that the elements of a set are definite, that no question arises as to whether a given object is a member of the set, is to say that the set is “well-defined”. I. Pronounce the following words: clarification simplification unification appreciate bind define unsuitable perception
[klærifi'keiwn] [simplifi'keiwn] [ja:nifi'keiwn] [w pri:LHLW@ [baind] [di'fain] [∧n'sjutwbl] [pw septwn]
notation intuitive refer qualify ambiguous avoid ambiguity distinguish
[nou'teiwn] [in'tjuitiv] [ri'fw:] ['kwolifai] [æm'bigjuws] [w void] [æmbi'gjuwli] [dis'tingwi@
18 II. Answer the following questions: 1. What promises clarification, simplification and unification in our teaching of mathematics? 2. Do the unifying principles bind certain ideas into a mathematical whole? 3. How could one choose to define “set”? 4. What is common practice? 5. What is a collection of items? 6. What is a set in our everyday speech? 7. How may the term “set” be restricted? 8. What does Cantor say about sets?
KhklZ\bl_evIjZ\rbg:e_dk_c
