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M1 Foundations

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T. Tsujioka

Advanced Seminar

M1 Foundations

Chapter 1. Basic Concepts

We start to introduce some basic concepts in mathematics in order to lay the solid foundations on which we build the edifice of mathematics step by step. Here, we freely use figures and diagrams to clarify the important points since it is harder for human mind to grasp the abstract concepts of algebra than concrete figures and diagrams; we can easily grasp the concepts when they are presented in figures and diagrams. We emphasize the point that there exist algebraic representations corresponding to geometric ones, and vice versa. Or, you may use the analytic point of view by introducing the coordinates to incorporate the both views. In physics, it is equivalent to establishing the observer’s point of view. You may note that you can find alternative solutions when you cannot find a solution from an algebraic point of view; that is, you can change a view from algebraic to geometric when you are at a loss in finding a solution, or vice versa. You should also note, however, that in spite of its abstruseness, the algebraic representations (expressions and rigorous rules) are precise in contrast with the geometrical representations. You would know that it is worth while acquiring those methods by making every effort.

1-1. The Concept of Set; Elements belonging to a Set; a Set included by another Set

To begin with, we need to introduce the concept of sets. A ‘set’ is a group of things. For example, students belonging to a school, items sold at a store, are considered to belong to an entire group.

Let’s denote a set of things be A. Things belonging to A are called elements.

When elements *a*, *b*, *c*,... belong to A, we denote as

.

When we want to specify a certain set consisting of elements *x*, we express as

When smaller set B exists and is included in A, we use the symbol or

and denote as (or whichever). When A = B, we attach = sign

underneath of the symbol as . The larger set which includes all the

smaller sets may be expressed as U (borrowed from the word Universe).

The symbol is usually used for the empty set that has no element. Use a diagram freely as in Fig. 1-1.

Several relationships with regard to sets are

Fig. 1-1

Symbols are used to represent the concepts concisely. When wiser symbols are

created, they may serve purpose of grasping the concepts better. But please note

that the symbols used in some textbooks may differ from those used in this

textbook. Always follow their definitions.
When you are confused with symbols, __always return to their definitions__; for everything begins with definitions in mathematics.

The following rules regarding 'intersection' and 'union' can be confirmed easily with the diagrams:

(1) Associative Law:

(2) Distributive Law:

The following relations, called **de Morgan’s rule**, may be confirmed also by use of the diagrams.

(3) (4)

Prob. 1.