The Language of Sets

1. Set theory is a language. Without it, not only can we do modern mathematics, we can’t even say what we are talking about. It is like trying to study French literature without knowing any French. The theory of sets, initiated by the German mathematician Cantor (1842-1918), constitutes the basis of almost all modern mathematics.

2. A set is often described as a collection of objects of any specified kind. However, such descriptions are no definitions as they merely replace the term “set” by other undefined terms. Thus the term “set” must be accepted as a primitive notion, without definition. The objects belonging to the set are the elements or members of the set. Although in introducing set theory it is helpful to work with concrete sets, whose members are real objects, the sets of interest in mathematics always have members which are abstract mathematical objects; the set of all circles in the plane, the set of points on a sphere, the set of all numbers.

3. We shall build up an algebra of sets. As in ordinary algebra, we shall use letters to represent sets and elements. We shall generally use small letters for elements and capital letters for sets, but it is impossible to keep rigidly to this convention because sets can themselves be elements of other sets.

4. A set is known if we know what its elements are. There are many ways of specifying a set, of which the simplest is to list all the members. The standard notation for this is to enclose the list in curly brackets. Two sets are equal if they have the same elements.

5. Instead of a list, we give a property which specifies precisely the elements we wish to be included in the set. If we are careful with our definitions this is as good as a list, and is usually more convenient. For sets with infinitely many members it is in any case impossible to give a complete list. The same is true for sets with a sufficiently large finite set of elements.

6. Sets with one element must not be confused with the element itself. It is not true that x and {x} are equal. It is confirmed by the observations that {x} has just one member, namely x, while x may have any number of members depending on whether or not it is a set, and if it is, which set.

Questions:

1. What is a set? 2. How are the objects of the set called? 2. In what case is a set considered to be known? 3. Which is the simplest way of specifying a set? 4. When are two sets equal? 5. What is the standard notation for a set? 5. What letters are generally used to represent elements of sets?

II. Перепишите и письменно переведите 2 и 5 абзацы текста.

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

1. Now we will be able to use transformation concept. 2. Must we consider the general principles of analytic continuation? – No, you needn’t. 3. Some of the quantities may be obtained as a result of the experiment. 4. If a more complicated mathematical function f(x) is to be calculated, one must express it in such a way that only four basic operations have to be applied. 5. Does he have to refer to this issue again? – No, he needn’t. 6. I failed to find the solution, I think I ought to try again.

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. A polynomial (are being made/is made/makes) up of several monomials. 2. The correctness of this assertion (is being verified/was being verified/had been verified) now. 3. Many of these functions (have been investigated/ investigated/had been investigated) in detail long before we started to work with them. 4. We shall avoid such complications if we (perform/shall perform/performed) only equivalent transformations. 5. Modern maths (began/was begun/begin) in ancient Greece. 6. The first systematic representation of calculus of finite differences (was given/gave/had been given) by Taylor in 1715.

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

1. This system always has a solution, since the determinant of its coefficients never vanishes because y' and y'' are linearly independent. 2. We have to perform some additional operations on the set of real numbers. 3. We shall have the results of the experiment published next week. 4. We have just distinguished the value b from the element b of Y. 5. The computer has been working for two hours since morning. 6. The value of this function has already been determined.

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

1. When substituting y and z into the equation, we finally evaluate x. 2. Given two points A and B, we can draw a line connecting them. 3. Having supposed the inequality, we obtained the necessary results. 4. Having been expressed in terms of symbols, these relations produced a formula. 5. The method applied in this case will give good results. 6. The methods being represented are the most important ones. 7. We saw him writing a report.

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

1. The problem having been stated, the students began solving it. 2. The speed of light being extremely great, we cannot measure it by ordinary methods. 3. In the solution of a quadratic equation all the terms are transposed to the left member, the right one being equal to zero. 4. Thus he considered a curve to be described by a moving point, the point being the point of intersection of two moving lines which were always parallel to two fixed lines at right angles. 5. We continued our work, with our laboratory assistants helping us. 6. With the experimental work completed, they could publish the results obtained.

VIII. Выберите правильный вариант и письменно переведите предложения.

1. Give that task to (nothing/somebody/anything) else. 2. Is there (somebody/anybody/everybody) to help you find the proof of the theorem? 3. You can’t find this book (somewhere/nowhere/anywhere), it is practically unavailable. 4. Using only a straightedge and a compass, the Greeks could easily divide (no/any/something) line segment into any number of equal parts. 5. (Nobody/Nowhere/No) can draw figures with such a high degree of accuracy. 6. Can you show me (some/no/anything) of your articles on this topic?