Grammar and vocabulary exercises

I. Read and translate the following sentences paying attention to “once” which is used as a conjunction.

1. Once we know what the answer is, we must verify it. 2. The proofs are not hard to understand once you see them. 3. Once some power becomes equal to1, the sequence must repeat. 4. Once we have chosen one set, we can use it instead of V. 5. Once I have got over this, there remains the problem of proving that the laws are true. 6. Once we do things this way, our condition is automatic. 7. Once we understand the idea, we can simplify our notation.

II. Read and translate the following sentences in which Participle I is used as an attribute.

1. The objects belonging to the set are the elements or members of the set. 2. S is a set consisting of a pencil and an eraser. 3. A line passing through a point inside a triangle must cut the triangle somewhere. 4. A line is the set of all pairs (x, y) satisfying its equation. 5. A set having operations of addition and multiplication which satisfy the nine laws of algebra is called a field. 6. The group of integers under addition has subgroups comprising all even integers. 7. The puzzle cannot be solved using lines that do not cross. 8. We get a new sequence containing every algebraic number. 9. An equation is a statement showing the equality of two quantities. 10. Fractions having different forms but equal values are equivalent fractions.

III. Read and translate the following sentences in which Participle I is used as an adverbial modifier.

1. Certain properties of the real world can be described using numbers. 2. Many of the concepts can be vividly illustrated using simple apparatus. 3. When using the curly bracket notation the elements are thought of as occurring once only in the set. 4. Proceeding in this way, we can set up the whole of Euclidean geometry as a part of set theory. 5. Having established that there is just one empty set, we can give it a symbol. 6. Having started from a system of axioms, we then could make certain logical deductions. 7There are many useful functions which are not easily defined using formulas. 8. Using algebra we can reduce complex problems to simple formulas. 9. When finding the product of multinomials we make use of the distributive law. 10. When speaking of quantities we shall have in view their numerical values.

IV. State the forms and the functions of Participle I in the following sentences and translate them.

1. The terms of an algebraic expression containing different letters are unlike terms. 2. Lobachevsky wrote a new geometry asserting that there could be several parallels. 3. This leads to a certain body of mathematics being thought of as “central”. 4. Having understood the theorem he could continue his calculations. 5. Having calculated the area we can say that the formula is exact. 6. Parallel lines are lines extending in the same direction and being the same distance apart no matter how far extended. 7. While finding answers to problems about the universe, the problems of building, cooking, measuring, buying and selling people use algebra. 8. Having understood the idea, we can simplify our notation. 9. If you use this notation, working in the group of integers under addition, then x y means x + y. 10. Having supposed an inequality we obtained the necessary results. 11. While discussing trigo- nometric functions of one of the acute angles of a right triangle, it is often useful to use a modification of the original definitions. 12. The scientists collecting information, formulating schemes need to express themselves in clear language. 13. Considering specific physical phenomena we may see that one and the same quantity in one phenomenon is a constant while in another it is a variable.

V. Ask questions to which the sentences below are the answers.

1. This operation assigns to any element x and y an element x * y belonging also to G. 2. Equations containing one or two variables to the first power only are linear in one or two variables. 3. This is the smallest field containing a given integral domain. 4. The student studying the theory of sets will find this statement interesting. 5. A point representing a variable is called a variable point. 6. Rational functions are functions involving an additional operation of division. 7. The group of integers under addition, has subgroups including all even integers, all multiples of 4, and so on. 8. Being proved by Lagrange, this theorem is widely used now. 9. Being interested in set theory he never missed his special courses. 10. Having understood the ideas we can simplify our notation. 11. There are really two types of problems involved here.

VI. Answer the following questions:

1. What branch of mathematics is the concept of a group abstracted from? 2. What does a group consist of? 3. What is the group operation required for? 4. What laws must the group operation satisfy? 5. In what situations can groups arise? 6. Do we have a group if one of the conditions fails? Why? Give an example and prove this. 7. When may the group operation be defined as a function? 8. When can we simplify the notation of a group?

VII. Translate into English.

Класс элементов а, в, с, … называется группой, если определена бинарная операция, которая каждой паре элементов а, в класса G ставит в соответствие некоторый элемент аов так, что:

1) а о в является элементом класса G;

2) ао (вос) = (а о в) о с (ассоциативный закон выполняется);

3) G содержит единицу Е такую, что для каждого элемента а из G,

Еоа = а;

4) для каждого элемента а из G в G существует обратный элемент а ^–1 такой, что а ^-1 оа = Е. Два элемента а, в, некоторой группы перестановочны, если аов = воа. Если все элементы а, в, группы G перестановочны, то определяющая операция называется коммутативной, а группа G – коммутативной или абелевой группой.