Basic Language of Mathematics

This book originates as an essential underlying component of a modern, imaginative three-semester honors program (six undergraduate courses) in Mathematical Studies. In its entirety, it covers Algebra, Geometry and Analysis in One Variable. The book is intended to provide a comprehensive and rigorous account of the concepts on sets, mapping, family, order, number (both natural and real), as well as such distinct procedures like Proof by Induction and Recursive Definition, and the interaction between these ideas; with attempts at including insightful notes on historic and cultural settings and.

151. The Real-Number System.

PREFACE; Some symbols; CONTENTS; Index of terms; Index of names; Index of conditions; Index of symbols; Chapter 1. SETS; 11. Introduction; 12. Sets and their members; 13. Inclusion; 14. Set formation; 15. Special sets; 16. Basic operations; 17. Pairs; product sets; 18. Partitions; Chapter 2. MAPPINGS; 21. The concept of a mapping; 22. The graph of a mapping; 23. The range of a mapping; images and pre-images; the partition of a mapping; 24. Inclusion, identity, and partition mappings; 25. Composition of mappings; diagrams; restrictions and adjustments; 26. Mappings from a set to itself.

Chapter 3. PROPERTIES OF MAPPINGS31. Constants; 32. Injective, surjective, and bijective mappings; 33. Inverses and invertibility; 34. Injectivity, surjectivity, and bijectivity: The induced mappings; 35. Cancellability; 36. Factorization; Chapter 4. FAMILIES; 41. The concept of a family; 42. Special families; 43. Families of sets; 44. Products and direct unions; 45. General associative and distributive laws; 46. Set-products and set-coproducts; Chapter 5. RELATIONS; 51. Relations in a set; 52. Images and pre-images; 53. Reversal, composition, and restriction of relations.

54. Relations from set t…