INTRODUCTION TO ABSTRACT MATHEMATICS

By Y. HINAE

CONTENTS WHAT IS MATHEMATICS AND WHAT DOES IT DO FOR US? IT’S NOT JUST NUMBERS MATHEMATICAL NOTATION MAKING THE INVISIBLE VISIBLE THIS IS WHERE YOU COME IN THE STUFF OF MODERN MATHEMATICS MATH SPEAK THE LANGUAGE OF MATHEMATICS: PART PROPERTIES OF THE LANGUAGE PROPERTIES OF QUANTI?CATION PROOFS IN MATHEMATICS THE INTEGERS MATHEMATICAL TRUTH SET THEORY SETS OPERATIONS ON SETS REAL INTERVALS ABSOLUTE VALUES INEQUALITIES ARBITRARY UNIONS AND INTERSECTIONS CARTESIAN PRODUCTS THE HISTORICAL DEVELOPMENT OF SET THEORY FUNCTIONS THE FUNCTION CONCEPT EXAMPLES OF FUNCTIONS HISTORY OF THE MODERN FUNCTION CONCEPT ONE-ONE AND ONTO FUNCTIONS COMPOSITION AND INVERSE FUNCTIONS DENUMERABILITY UNCOUNTABILITY RELATIONS BINARY RELATIONS PROPERTIES OF RELATIONS RELATIONS AS SETS OF ORDERED PAIRS RELATIONS AS GRAPHS EQUIVALENCE RELATIONS FUNCTIONS AS RELATIONS AN EXAMPLE: THE REALS UPPER BOUNDS COMPLETENESS SEQUENCES NO ANSWERS TO THE EXERCISES