Basic Concept of Mathematics

Contents:
PrefaceAbout the Author
Chapter 1. Some Set Theoretical Notions
1. Introduction. Sets and their Elements
2. Operations on Sets, Problems in Set Theory
3. Logical Quantiers
4. Relations (Correspondences), Problems in the Theory of Relations
5. Mappings, Problems on Mappings
6. Composition of Relations and Mappings, Problems on the Composition of Relations
7. Equivalence Relations, Problems on Equivalence Relations
8. Sequences, Problems on Sequences
9. Some Theorems on Countable Sets, Problems on Countable and Uncountable Sets
Chapter 2. The Real Number System
1. Introduction
2. Axioms of an Ordered Field
3. Arithmetic Operations in a Field
4. Inequalities in an Ordered Field. Absolute Values, Problems on Arithmetic Operations and Inequalities in a Field
5. Natural Numbers. Induction
6. Induction (continued), Problems on Natural Numbers and Induction
7. Integers and Rationals, Problems on Integers and Rationals
8. Bounded Sets in an Ordered Field
9. The Completeness Axiom. Suprema and Infima, Problems on Bounded Sets, Infima, and Suprema
10. Some Applications of the Completeness Axiom, Problems on Complete and Archimedean Fields
11. Roots. Irrational Numbers, Problems on Roots and Irrationals
12. Powers with Arbitrary Real Exponents, Problems on Powers
13. Decimal and other Approximations, Problems on Decimal and q-ary Approximations
14. Isomorphism of Complete Ordered Fields, Problems on Isomorphisms
15. Dedekind Cuts. Construction of E1, Problems on Dedekind Cuts
16. The Infinities. The lim and lim of a Sequence, Problems on Upper and Lower Limits of Sequences in E
Chapter 3. The Geometry of n Dimensions.
1. Euclidean n-space, En Problems on Vectors in En
2. Inner Products. Absolute Values. Distances Problems on Vectors in En (continued)
3. Angles and Directions
4. Lines and Line Segments, Problems on Lines, Angles, and Directions in En
5. Hyperplanes in En, Linear Functionals on En, Problems on Hyperplanes in En
6. Review Problems on Planes and Lines in E3
7. Intervals in En, Additivity of their Volume, Problems on Intervals in En
8. Complex Numbers, Problems on Complex Numbers
9. Vector Spaces. The Space Cn, Euclidean Spaces, Problems on Linear Spaces
10. Normed Linear Spaces, Problems on Normed Linear Spaces
Notation
Index
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