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I fully agree to it. Not quite. It’s unlikely.
I don’t think this is the case. Just the reverse.
1. In ordinary algebra we use small letters to represent sets and capital letters to represent elements. 2. Sets can themselves be elements of other sets. 3. Two sets are equal if they have different elements. 4. The order inside the brackets makes difference. 5. A set is considered to be known if we know what its elements are. 6. The mathematical notion of a set doesn’t allow sets with one member. 7. The empty set is “nothing”. 8. The empty set must not be confused with the number 0.
Ex. 12. Read the following sentences, find the Absolute Participle Construction and translate them into Russian.
1. Each major concept embraces not one but many diverse objects, all having some common property. 2. The theorem having been stated, the students began proving it. 3. We may use two different methods, the first being the more general one. 4. This system consists only of one equation, the other two being its consequences. 5. The theorem being true, we cannot assume that its converse must be true. 6. A function being continuous at every point of the set, it is continuous throughout the set. 7. No sign preceding a term, the plus sign is understood. 8. The coordinates being given, we can specify the position of any point in the plane. 9. The set is bounded above and below, there being numbers greater than and numbers smaller than all the numbers in the set.
Ex. 13. Read and translate the following sentences paying attention to the translation of ONE as the subject.
Model: One must … Нужно …
One can … Можно …
One should … Следует …
One needs … Необходимо …
One knows … Знаешь …
1. Another place where one must be careful about logic is when proving something impossible. 3. In elementary mathematics one comes across various objects designated by the term “function”. 4. One must learn how to draw graphs. 5. In this case one needs to consider all possible proofs. 6. Similarly one can prove the other laws of arithmetic. 7. One should not be surprised at this. 8. In order to apply group theory to a branch of mathematics, one must check that the relevant objects are groups. 9. For practical purposes one needs good approximate constructions. 10. One must realize that the development of mathematics was by no means the product of one individual’s efforts.
Ex. 14. Say this in English.
1. Множество – это набор каких-либо объектов, называемых его элементами и обладающих общим для всех них характерным свойством. 2. Обычно множества обозначаются заглавными буквами, а члены множества строчными буквами. 3. Объекты, которые принадлежат множеству, являются элементами или членами множества. 4. Множества, представляющие интерес в математике, содержат абстрактные математические объекты. 5. Считают, что множество известно, если мы знаем его элементы. 6. Существует много способов точного определения множества. 7. Два множества равны, если они содержат одинаковые элементы. 8. Множества с одним элементом не нужно путать с самим элементом. 9. Множество, не содержащее элементы, называется пустым. 10. Пустое множество нельзя путать с нулём, так как 0 – это число, а пустое множество – это множество.
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Ex. 15. Topics for discussion:
1. Give the definition and properties of a set.
2. Dwell on the sets in algebra.
3. Describe sets in everyday life.
4. Give the ways of specifying a set.
5. Speak on the notion of an empty set.
Ex. 16. Read the text and find the answers to the following questions.
1. What does set theory study?
2. What objects is set theory applied to?
3. Where can the language of set theory be used?
4. When was the modern study of set theory initiated?
5. What does set theory begin with?
6. What is the subset relation?
7. How many full derivations of complex mathematical theorems from set theory have been formally verified?
TEXT B
SET THEORY
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by George Cantor and Richard Dedekind in the 1870s.
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or an element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3}, but {1,4} is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule out this, the term ‘proper subset’ is defined. A is called a proper subset of B if and only if A is a subset of B, but B is not a subset of A.
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present.
Set theory is a major area of research in mathematics, with many interrelated subfields.
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UNIT III
Grammar: THE GERUND. ITS FORMS AND FUNCTIONS .
FORMS OF THE GERUND
| INDEFINITE | PERFECT | |
| ACTIVE | asking writing | having asked having written |
| PASSIVE | being asked being written | having been asked having been written |
Ex. 1. Study the functions of the Gerund. State their forms. Translate the sentences into Russian.
a) the Gerund is used as a Subject.
1. Writing a sentence in algebraic form, as we have seen, involves two steps. 2. Drawing a straight line in one direction gives you a one way extension. 3. Reducing a fraction means bringing it to lower terms. 4. Reading slowly is useful for beginners. 5. Locating the point on the y-axis gives you the first point on the line. 6. Knowing the properties of equality will help you decide whether a sentence is true or false.
b) the Gerund is used as a part of a Predicate.
1. Our task is proving the correctness of the given statement. 2. The young scientist began experimenting. 3. We expected being given further assistance. 4. This terminology needs improving. 5. The scientist expected being included in the experimental group. 6. She stopped investigating the problem as her approach was wrong.
c) the Gerund is used as a Direct object.
1. We discussed improving the shape of the model. 2. Do you mind being examined first? 3. I don’t remember speaking to him about this fact. 4. Avoid making such bad mistakes. 5. They are busy now reading the text. 6. He suggested taking part in this conference.
d) the Gerund is used as a Prepositional object.
1. He was prevented from finishing his work. 2. We succeeded in accomplishing our task. 3. He insisted on writing the thesis as soon as possible. 4. These computers are capable of solving systems with a hundred or more unknowns, if necessary. 5. They are concerned with applying their knowledge of the subject to solving these problems. 6. We cannot agree to testing the new method without being given additional time.
e) the Gerund is used as an Attribute.
1. What ways of learning words do you find most effective? 2. This is the method of doing such tasks. 3. I can’t improve my English because I don’t have any opportunities of speaking it. 4. The idea of using symbols instead of words proved very helpful. 5. There exists a very efficient algorithm for solving most linear programming problems. 6. The procedure of reducing a fraction to its lowest terms is not complicated.
f) the Gerund is used as an Adverbial modifier.
1. In considering the problem we have to deal with the laws of motion. 2. The product may be found by multiplying the factors contained in the given mathematical sentence. 3. We can’t agree to testing the new method without being given additional time. 4. In naming geometric objects we often use capital letters. 5. By applying the knowledge of geometry you can locate the point in the plane. 6. After discussing the problem in detail they found the best solution.
Ex. 2. State the form and the function of the Gerund. Translate the sentences into Russian.
1. We insisted on carrying out another experiment to check the results.
2. The absolutely new contribution made by Descartes was in importing the idea of motion into geometry.
3. This is the basic method of solving problems of statics.
4. It is worth noting that the work of the early Arab mathematicians makes no clear division between arithmetic and algebra.
5. Since the equation is linear and has constant coefficients it can be easily solved by using classical differential equation theory.
6. He also improved the notation for representing the extraction of roots.
7. Combining the integrals gives the following equation.
8. The preceding definitions have laid the foundation for considering the variation of a functional.
9. Leonardo’s solution is worth quoting for its elegance.
Ex. 3. Put the Gerund in the correct form. Use prepositions where necessary.
1. She continued (to translate) the text from English into Russian.
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2. He found the product (to multiply) the numerals.
3. She is afraid (to take) the exam.
4. He used a ruler (to draw) a straight line.
5. One must be very careful (to measure) the volume of an object.
6. They insist on the question (to reconsider).
7. She stopped (to investigate) the problem as her approach was wrong.
8. The problem (to discuss) various points of view was a very useful exercise.
9. Measurement is a process (to associate) numbers with certain objects.
Ex. 4. Insert prepositions (in, of, to, from, by). You can use the same preposition in more than one sentence.
1. His mathematical power, which never failed him to the end of his life, was employed at this period __ originating the calculus of probabilities, and __ inventing the arithmetical triangle.
2. The mathematician who came nearest __ solving the challenge questions issued by Pascal on the cycloid (циклоид) was John Wallis.
3. But he differs from B. F. Cavalieri (an Italian mathematician) __ regarding lines as made up of infinitely small lines, surfaces of infinitely small surfaces, and volumes of infinitely small volumes.
4. Leonardo’s favorite method __ solving many problems is by the method of ‘false assumption’, which consists __ assuming a solution and then altering (изменение) it by simple proportion as in the rule of three (вычислительный метод в математике).
5. We have succeeded __ verifying that the increment (приращение) can be written in the form of the following equation.
6. His famous experiment __ dropping bodies of different weights from the tower of Pisa enabled him to demonstrate that all bodies undergo the same acceleration __ falling towards the earth, a result which his experiment with light and heavy pendulums (маятники) also proved.
7. He also discusses solids generated __ revolving a curve about an axis, and in the last section deals with the problems of maxima and minima.
8. The intellectual trend of that time was such as to prevent mathematics __ becoming a popular subject.
9. Since Euler’s equations usually cannot be solved analytically, one naturally thinks __ using numerical integration.
10. F. Viet (a French mathematician) succeeded __ finding 23 of the 45 roots.
Ex. 5. Change the time clause into the ‘in + Gerund’ structure.
Model: He made a mistake when he was proving the theorem.
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