Grammar and vocabulary exercises. I. Give the Russian equivalents for

I. Give the Russian equivalents for:

in introducing set theory, the sets of interest in mathematics, abstract mathematical objects, to keep rigidly to this convention, at any rate, many ways of specifying a set, to list all the members, to enclose the list in curly brackets, the curly bracket notation, elements listed more than once, inside the brackets, it makes no difference, instead of a list, infinitely many members, none at all, to tell them apart.

II. Find the English equivalents of the following Russian word combinations:

1) члены множества; 2) теория множеств; 3) конкретные множества; 4) абстрактные объекты; 5) способ определять множество; 6) обычно принятое обозначение; 7) фигурные скобки; 8) одни и те же элементы; 9) перечисленные элементы; 10) точное свойство; 11) бесконечное число элементов; 12) полный перечень; 13) конечное число элементов 14) толь- ко один элемент.

a. extract property, b. curly brackets, c. members of the set, d. set theory, e. just one element, f. a finite set of elements, g. elements listed, h. concrete sets, i. a way of specifying a set, 1g. the same elements, k. infinitely many elements, l. standard notation, m. abstract object, n. a complete list.

III. Insert the suitable words: (empty, members, to list, notation, specifying, difference, precisely, the same)

1.The objects belonging to the set are the elements or … of the set. 2. There are many ways of … a set. 3. The simplest way of specifying a set is … all the members. 4. The standard … is to enclose the list in curly brackets. 5. Two sets are equal if they have … elements. 6. The order inside the brackets makes no …. 7. Instead of a list, we give a property which specifies … the elements. 8. A set with no element is called an … set.

IV. Ask questions for which the given sentences are answers.

1. A set is a collection of objects. 2. The objects belonging to the set are the elements or members of the set. 3. The sets of interest in mathematics always have members which are abstract mathematical objects. 4. In the algebra of sets we use letters to represent sets and elements. 5. A set is considered to be known if we know what its elements are. 6. There are many ways of specifying a set. 7. The standard notation is to enclose the list in curly brackets. 8. Two sets are equal if they have the same elements. 9. Instead of a list, we give a property which specifies precisely the elements of the set. 10. For sets with infinitely many members, it is impossible to give a complete list. 11. The mathematical notion of a set allows sets with only one member – or even no members at all. 12. A set with no elements is called an empty set. 13. All empty sets are equal.

V. 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 solving 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 must not 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. 10. A limit existing, it represents the instantaneous rate of change of f(x) at the point x.

VI. Read and translate the following sentences paying attention to the translation of one as the subject.

1. One is always uncomfortable when faced with the unknown. 2. 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.

VII. Answer the following questions:

1. What is a set? 2. What are the elements of the set? 3. What sets are of interest in mathematics? 4. What do we use to represent sets and elements? 5. What set is considered to be known? 6. What is the simplest way of specifying a set? 7. What is the standard notation for a set? 8. What sets are equal? 9. How can we specify the elements of a set? 10. How many members may a set have? 11. What is an empty set? 12. How is an empty set represented? 13. Does an empty set exist at all?