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REAL ANALYSIS [For B.A., B.Sc. and Honours (Mathematics and Physics), M.A. and M.Sc. (Mathematics) students of various Universities/Institutions as per UGC Model Carriculum. Also useful for GATE and various other competitive examination

S. CHAND & COMPANY LTD. (AN ISO 9001: 2008 COMPANY) RAM NAGAR, NEW DELHI-110 055   1. SETS AND FUNCTIONS ... 1.1–1.9 1.1. Introduction ... 1.1 1.2. Statements ... 1.1 1.3. Connectives ... 1.2 1.4. Sets ... 1.4 1.5. Functions (or mappings) ... 1.6 1.6. Composite of functions (or product of functions) ... 1.7 1.7. Inverse function ... 1.8 1.8. Binary operation ... 1.8 Objective questions ... 1.9 2. THE REAL NUMBERS ... 2.1–2.32 2.1. Introduction ... 2.1 2.2. The set N of natural numbers ... 2.2 2.3. The set I or Z of integers ... 2.3 2.4. The set Q of rational numbers ... 2.4 2.5. The set of real numbers R as a complete ordered field ... 2.6 2.6. Closed, open, semi-closed and semi-open intervals ... 2.7 2.7. Set bounded above, set bounded below, l.u.b. (supremum) and g.l.b. (infimum) of a set. The greatest and smallest members of a set. Bounded and unbounded sets ... 2.7 2.8. Order-completeness of the set of real numbers ... 2.16 2.9. Equivalent descriptions of the order-completeness. Property of the set of real numbers ... 2.18 2.10. Explicit statement of the properties of the set of real numbers as a complete ordered field ... 2.20 2.11. Some important properties of the system of real numbers ... 2.20 2.12. The denseness property of the set of real numbers R ... 2.21 2.13. The modulus (or absolute value) of a real number ... 2.26 2.14. Arithmetic and geometric continua ... 2.29 Objective questions ... 2.31 3. NEIGHBOURHOODS AND LIMIT POINTS OF A SET. OPEN AND CLOSED SETS ... 3.1–3.36 3.1. Introduction ... 3.1 3.2. Neighbourhood of a point ... 3.1 3.3. Properties of neighbourhoods ... 3.3 3.4. Limit (or accumulation or condensation) point of a set ... 3.4 3.5. Existence of limit points ... 3.8 3.6. Open and closed sets ... 3.13 3.7. Basic theorems concerning families of open and closed sets ... 3.15 3.8. Illustrations of open sets ... 3.18 3.9. Illustrations of closed sets ... 3.19 3.10. Interior point and interior of a set ... 3.22 3.11. Exterior point and exterior of a set ... 3.22 3.12. Boundary (or frontier) point and boundary (or frontier) of a set ... 3.22 3.13. Theorems on interior of a set ... 3.23 3.14. Adherent point (or a contact point) and closure of a set ... 3.25 3.15. Theorems on closure of a set ... 3.26 (iv) Created with Print2PDF. To remove this line, buy a license at: h

 301. SETS AND FUNCTIONS ... 1.1–1.9 1.1. Introduction ... 1.1 1.2. Statements ... 1.1 1.3. Connectives ... 1.2 1.4. Sets ... 1.4 1.5. Functions (or mappings) ... 1.6 1.6. Composite of functions (or product of functions) ... 1.7 1.7. Inverse function ... 1.8 1.8. Binary operation ... 1.8 Objective questions ... 1.9 2. THE REAL NUMBERS ... 2.1–2.32 2.1. Introduction ... 2.1 2.2. The set N of natural numbers ... 2.2 2.3. The set I or Z of integers ... 2.3 2.4. The set Q of rational numbers ... 2.4 2.5. The set of real numbers R as a complete ordered field ... 2.6 2.6. Closed, open, semi-closed and semi-open intervals ... 2.7 2.7. Set bounded above, set bounded below, l.u.b. (supremum) and g.l.b. (infimum) of a set. The greatest and smallest members of a set. Bounded and unbounded sets ... 2.7 2.8. Order-completeness of the set of real numbers ... 2.16 2.9. Equivalent descriptions of the order-completeness. Property of the set of real numbers ... 2.18 2.10. Explicit statement of the properties of the set of real numbers as a complete ordered field ... 2.20 2.11. Some important properties of the system of real numbers ... 2.20 2.12. The denseness property of the set of real numbers R ... 2.21 2.13. The modulus (or absolute value) of a real number ... 2.26 2.14. Arithmetic and geometric continua ... 2.29 Objective questions ... 2.31 3. NEIGHBOURHOODS AND LIMIT POINTS OF A SET. OPEN AND CLOSED SETS ... 3.1–3.36 3.1. Introduction ... 3.1 3.2. Neighbourhood of a point ... 3.1 3.3. Properties of neighbourhoods ... 3.3 3.4. Limit (or accumulation or condensation) point of a set ... 3.4 3.5. Existence of limit points ... 3.8 3.6. Open and closed sets ... 3.13 3.7. Basic theorems concerning families of open and closed sets ... 3.15 3.8. Illustrations of open sets ... 3.18 3.9. Illustrations of closed sets ... 3.19 3.10. Interior point and interior of a set ... 3.22 3.11. Exterior point and exterior of a set ... 3.22 3.12. Boundary (or frontier) point and boundary (or frontier) of a set ... 3.22 3.13. Theorems on interior of a set ... 3.23 3.14. Adherent point (or a contact point) and closure of a set ... 3.25 3.15. Theorems on closure of a set ... 3.26 (iv) Created with Print2PDF. To remove this line, buy a license at: h3