Speech exercises. Often one set is part of some other set

I. Read and reproduce.

SUBSETS

Often one set is part of some other set. The set of all women is part of the set of all human beings; the set of all even numbers is part of the set of all whole numbers. Again the phrase “part of” has unfortunate connotations, and mathematicians have been forced to invent a new word to denote the precise concept involved.

A set S is said to be a subset of a set T provided that every member of S is a member of T. Every member of the set W of all women is a woman, hence a human being, hence a member of the set H of all human beings. So W is a subset of H. We use the notation

W H

to describe this, so the symbol should be read as “is a subset of”. We also say that W is contained in H. Certain facts follow at once from our definition of “subset”. Every set is a subset of itself – because all of its members are members of it. Further, the empty set Ø is a subset of any set you care to name. If it were not a subset of a given set S, then there would have to be some element of Ø which was not an element of S. In particular there would have to be an element of Ø. Since Ø has no elements this is impossible.

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 B is an element of C, then every element of A is an element of C.

II. Topics for discussion:

1. Often one set is part of some other set.

2. Every set is a subset of itself.

3. A subset of a subset is itself a subset.

III. Say this in English.

Множество – это набор, совокупность, собрание каких-либо объектов, называемых его элементами и обладающих общим для всех их характеристическим свойством. “Множество есть многое, мыслимое нами как единое” (Кантор). Это не является в полном смысле логическим определением понятия множества, а всего лишь пояснением (так как определить понятие – значит найти такое родовое понятие, в которое данное понятие входит в качестве вида, но множество – это, пожалуй, самое широкое понятие математики и логики). При этом можно либо дать перечень элементов множества, либо правило, по которому можно определить, принадлежит или нет данный объект рассматриваемому множеству. Это правило часто называют характеристическим свойством множества.

T E X T III

ORDINARY DIFFERENTIAL EQUATIONS

GRAMMAR: THE GERUND