I fully agree to it. Not quite. It’s unlikely.

I don’t think this is the case. Just the reverse.

1. In ordinary algebra we use small letters to represent sets and capital letters to represent elements. 2. Sets can themselves be elements of other sets. 3. Two sets are equal if they have different elements. 4. The order inside the brackets makes difference. 5. A set is considered to be known if we know what its elements are. 6. The mathematical notion of a set doesn’t allow sets with one member. 7. The empty set is “nothing”. 8. The empty set must not be confused with the number 0.

Ex. 12. 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 proving 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 cannot 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.

Ex. 13. Read and translate the following sentences paying attention to the translation of ONE as the subject.

Model: One must … Нужно …

One can … Можно …

One should … Следует …

One needs … Необходимо …

One knows … Знаешь …

1. 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.

Ex. 14. Say this in English.

1. Множество – это набор каких-либо объектов, называемых его элементами и обладающих общим для всех них характерным свойством. 2. Обычно множества обозначаются заглавными буквами, а члены множества строчными буквами. 3. Объекты, которые принадлежат множеству, являются элементами или членами множества. 4. Множества, представляющие интерес в математике, содержат абстрактные математические объекты. 5. Считают, что множество известно, если мы знаем его элементы. 6. Существует много способов точного определения множества. 7. Два множества равны, если они содержат одинаковые элементы. 8. Множества с одним элементом не нужно путать с самим элементом. 9. Множество, не содержащее элементы, называется пустым. 10. Пустое множество нельзя путать с нулём, так как 0 – это число, а пустое множество – это множество.

Ex. 15. Topics for discussion:

1. Give the definition and properties of a set.

2. Dwell on the sets in algebra.

3. Describe sets in everyday life.

4. Give the ways of specifying a set.

5. Speak on the notion of an empty set.

Ex. 16. Read the text and find the answers to the following questions.

1. What does set theory study?

2. What objects is set theory applied to?

3. Where can the language of set theory be used?

4. When was the modern study of set theory initiated?

5. What does set theory begin with?

6. What is the subset relation?

7. How many full derivations of complex mathematical theorems from set theory have been formally verified?