The Empty Set and Subsets

1. A set with no elements is called an empty set. Instead of saying that there are no objects of some specific kind, the set of these elements is said to be empty; however this set itself, though empty, will be regarded as an existing thing.

2. All empty sets are equal. Recall that two sets are equal if they have the same members. If they are unequal, then they do not have the same members, sо one of then must have at least one member that the other does not have. In particular, one of them must have a member. If they are both empty, this is not the case; sо they are unequal. Therefore they must be equal.

3. The empty set is not "nothing". It is just as much in existence as any other set. It is its members that do not exist. It must not be confused with the number 0: for 0 is a number whereas Ø is a set. Ø is one of the most useful sets in mathematics. One of its uses is to express concisely that something does not happen.

4. Once a set has been formed, it is regarded as a new entity, that is, a new object, different from any of its elements. This object may, in its turn, be an element of some other set. For example, the set of all even numbers is part of the set of all whole numbers. A set S is called a subset of a set T provided that every member of S is a member of T.

5. Every set is a subset of itself, because all of its members are members of it. One nice property of subsets is that a subset of a subset is itself a subset: if A B and B C then A C. For if every element of A is an element of B, and if every element of В is an element of C, then every element of A is an element of С.

6. Sets may be combined together to give other sets. Prominent among the infinite number of possible ways of combining them are the union and intersection of sets. The union of two sets S and T is the set whose elements are those of S, together with those of T. We use the symbols S U T. In a similar fashion the intersection S ∩ T is the set whose members are the elements which are common to S and T. The symbols U and ∩ obey various general laws in the same way that addition and multiplication of numbers obey certain general laws.

Questions:

1. What is an empty set? 2. When are two sets equal? 3. What is the difference between 0 and an empty set? 4. In what case is the set S called a subset of the set T? 5. What is one of the properties of subsets?

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

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

1. I am able to make a drawing of this figure. 2. In the discussion of these problems two stages must be distinguished. 3. A parabola may be regarded as an ellipse with one of its foci removed into infinity. 4. It is to be noted that this procedure applies only to interior points of the interval. 5. Do we have to define conic sections? – No, you needn’t. 6. The text should be read and reread as many times at home as it is necessary for every student to grasp the meaning.

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. Polynomials (were studied/had been studied/was studied) at the previous lesson. 2. The sum of sets of solutions of these systems (will consider/will have considered/will be considered) later on. 3. The theoretical character of the issue (was being discussed/ is being discussed/is discussing) now. 4. It was pointed out that equations of the n-th order (had been introduced/ introduced/have been introduced) in a similar way earlier. 5. Riemann (was introduced/ introduced/introduce) a more general kind of geometry. 6. These propositions (require/are required/will be required) proofs which will be omitted here.

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

1. His diagram has certain advantages. 2. One often has to establish what kinds of solutions exist. 3. We must have all the equipment packed by tomorrow. 4. We have just represented any real number either exactly or approximately by using the point. 5. I haven’t been doing any research since last year. 6. Have these fractions been reduced?

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

1. Adding fractions in arithmetic, you must determine the least common denominator of the fractions involved. 2. The function f belonging to one of these classes depends on one variable x and can be real or complex. 3. If changed a little, the problem will be easy to solve. 4. Having performed the necessary operations, they obtained the differential equation of the second order. 5. Having been disappointed by the results, we changed the lines of our research. 6. The right member of the equation being considered contained no unknowns. 7. I heard them discussing the term “ if and only if ”.

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

1. The theorem being true, we must not assume that its converse must be true. 2. All the necessary changes having been made, the experiment showed different results. 3. We may use two different methods, the first being the more general one. 4. A simple example of his actual method is given by his proof that a parallelogram is divided by its diagonal into two triangles, each having half the area of the parallelogram. 5. With an object moving at constant speed, the distance covered is directly proportional to time. 6. We have drawn a triangle, with the measure of its altitude being three times the measure of its base.

VIII. Выберите правильный вариант и письменно переведите предложения.

1. (Some/Any/No) notions cannot be defined in a precise and explicit way. 2. Did he find (something/anything/anywhere) as a result of this operation? 3. They needn’t perform (any/no/some) other construction. 4. You may come at (no/any/something) time that is convenient to you. 5. (No/No one/Nobody) mathematician confuses these basic terms. 6. Can I take (any/some/nothing) of these books on topology?