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 а о в так, что:

1) а о в является элементом класса G;

2) а о (в о с) = (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^-1оа = 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.