Контрольнoe задание № 1

I. Прочитайте текст, переведите его и письменно ответьте на вопросы, следующие за текстом.

Sets

1. The concept of a set of objects or elements is outwardly simple and has an intuitive counterpart in everyday experience. Although the mathematician in formally developing mathematical systems would prefer to consider the term «set» to be undefined, it is easy to suggest clearly both the meaning and the use of the term. A set is any well-defined collection of distinct objects, and it is synonymous with the terms aggregate or class.

2. Although this concept is quite general, one must be careful to observe two things: a set consists of distinct elements of objects, so that every element of a set is a separate object, and no two elements of a set are identical.

3. An element will always be considered as a member (or object) of some set; elements will not be permitted to exist without a parent set. It is evident that stating the property possessed by the elements of a set is the most meaningful way to specify a set. Mathematicians are free to consider sets consisting of a finite number of elements, sets containing infinitely many elements or even sets consisting of only one element, say the set {a}. Indeed, one-element sets are very important, for the set {a} is not the same thing as the element a. Furthermore, the elements of sets may be any definite objects whatever and they may themselves be sets (in which case the set will be called «a collection of sets» rather than a set of sets). The «empty set» or the set which consists of no elements will also be admitted into the family sets. Although it may appear strange to admit the exis­tence of the «empty or null set», this concept is not as artificial as it might first appear to be.

Operations with Sets

4. Two elements are equal, x=y, if x and у are the same element. Thus the word «equals» in mathematics means that the elements x and y are two names for the same object. This leads to a definition for the equality of two sets: The sets А, В are equal if they contain the same elements. If we are given a set, it seems natural to think of subdividing the set and to consider parts of it. Parts of a set are called subsets. The set R is a subset of the set T if every element of R is an element of T.

5. The Union of the sets A and В is the set of all elements which belong either to A or to В or to both A and B.

6. The product (or intersection) of the sets A and В is the set of elements which belong to both A and B.

7. The difference В — A is the subset of В whose elements are not in A.

It is required that an element should be a member of some set, and all sets under discussion should be subsets of some given (or universal) set.

Questions:

1. What is a set? 2. What is the most meaningful way to specify a set? 3. What is an empty set? 4. When are two sets equal? 5. In what case is the set R called a subset of the set T? 6. Why are one-element sets very important?

II. Перепишите и письменно переведите 1 и 3 абзацы текста.

III. Переведите письменно предложения, обращая внимание на перевод модальных глаголов.

1. We were able to express this algebraically by taking coordinates in the Euclidian plani. 2. Must we write the corresponding relations? - No, you needn’t. 3. This system may be solved in another way without reducing it to an equation of the n-th order. 4. It is to be expected that the axioms which we obtain will be very intuitive. 5. Do they have to reverse their approach to the problem? - No, they needn’t. 6. He should specify the conditions of the experiment.

IV. Выберите правильный вариант формы глагола в страдательном или в действительном залоге. Переведите письменно предложения, учитывая способы перевода страдательного залога и порядок слов в русском и английском языках.

1. This method (will be followed/will follow/will have followed) in all cases. 2. By equivalent transformations every linear equation with one variable (brought/was brought/was bringing) to this general form. 3. The set theory (is being discussed/discusses/is discussing) now. 4. Before an algebraic symbolic language (has been developed/had been developed/had developed), equations were written in words. 5. We sometimes (encounter/are encountered/is encountering) integrals whose limits of integration can dependon a parameter. 6. The early geometers (deals/dealt/were dealt) with measurements of line segments, angles and other figures in a plane.

V. Переведите письменно предложения, обращая внимание на различные значения глагола to have.

1. Two non-parallel planes always have a line in common which is called a line of intersection. 2. The existence or nonexistence of the proof had to be developed. 3. You should have your course paper checked by your science advisor. 4. From the geometrical point of view we have obtained the totality of all the possible straight lines passing through the origin of coordinates. 5. We had been discussing the problem for two hours when you came. 6. The circumference of a circle has not always been clearly defined.

VI. Переведите письменно предложения, принимая во внимание особенности перевода на русский язык причастий и причастных оборотов.

1. While discussing trigonometric functions of one of the acute angles of a right triangle, it is often helpful to use a modification of the original definitions. 2. When applied carefully, this method may prove to be very useful. 3. Having obtained the necessary devices, we could finish our experiment. 4. Having been read the book was taken back to the library. 5. Drawing a picture of two intersecting lines the student understands better the idea of the interior and exterior of the angles obtained. 6. The set being considered contains no elements. 7. I saw her working at her thesis.

VII. Переведите письменно предложения, содержащие независимые причастные обороты, учитывая при переводеих место в предложении.

1. The values of the unknowns being very large, we cannot calculate the equation by ordinary methods. 2. The measure of the circumference given, you can find the area of the interior of the circle. 3. Approximate integration methods are discussed in Chapter 5, their significance being increased by the fact that integrable systems occur so rarely in reality. 4. This system consists only of one equation, the other being its consequence. 5. With the distance having been defined, you can expect to find the speed. 6. The result can also be denoted as in the figure given above, with force F neglected.

VIII. Выберите правильный вариант и письменно переведите предложения.

1. There are (any/some / nothing) problems that are impossible to solve. 2. Are there (some/any / anything) equations in this text? 3. He doesn’t know (nothing/anything / something) about this method. 4. You may take (nothing/anything / somewhere) you like. 5. (Nobody / Nothing/Anything) can perform the drawing in a better way. 6. Can you present (some/nothing/something) proofs of the theorem?