FUNDAMENTAL CONCEPTS OF MATHEMATICS
CHAPTER 1 SET INTRODUCTION BASIC DEFINITION OF SET SET NOTATION DIFFERENT NOTATIONS IN SETS TYPES OF SETS SET OPERATIONS VENN DIAGRAMS VENN DIAGRAM HISTORY VENN DIAGRAM PURPOSE AND BENEFITS ALGEBRA OF SETS THE FUNDAMENTAL LAWS OF SET ALGEBRA THE PRINCIPLE OF DUALITY SOME ADDITIONAL LAWS FOR UNIONS AND INTERSECTIONS SOME ADDITIONAL LAWS FOR COMPLEMENTS THE ALGEBRA OF INCLUSION THE ALGEBRA OF RELATIVE COMPLEMENTS AXIOMATIC SET THEORY SCHEMAS FOR GENERATING WELL-FORMED FORMULAS AXIOMS FOR COMPOUNDING SETS AXIOMS FOR INFINITE AND ORDERED SETS SCHEMA FOR TRANSFINITE INDUCTION AND ORDINAL ARITHMETIC AXIOM FOR ELIMINATING INFINITE DESCENDING SPECIES CARDINALITY SUMMARY MULTIPLE CHOICE QUESTIONS REVIEW QUESTIONS REFERENCES STATEMENTS, LOGICAL CONNECTIVES VALIDITY, TRUTH TABLE TAUTOLOGIES QUANTIFIERS THE UNIVERSAL QUANTIFIER UNIVERSAL QUANTIFIER AND CONNECTIVE AND THE EXISTENTIAL QUANTIFIER METHODS OF PROOF TRIVIAL PROOF/VACUOUS PROOF DIRECT PROOF PROOF BY CONTRAPOSITIVE PROOF BY CONTRADICTION PROOF BY CASES EXISTENCE PROOFS CONSTRUCTIVE PROOF NONCONSTRUCTIVE PROOF NONEXISTENCE PROOFS THE HALTING PROBLEM COUNTEREXAMPLE COUNTEREXAMPLE TO ?XP(X) BICONDITIONAL PROOF STRATEGIES CHAPTER 3 RELATION AND FUNCTION RELATION DEFINITION EQUIVALENCE RELATIONS EQUIVALENCE CLASSES AND PARTITIONING PARTIAL ORDERING SOLUTION FUNCTION INJECTION SURJECTION AND BIJECTION IMAGE AND INVERSE IMAGE INVERSE FUNCTION CARDINAL NUMBER OF A SET COUNTING PRINCIPLES COUNTABLE AND UNCOUNTABLE SETS UNCOUNTABLE SETS CHAPTER 4 INTEGERS CONCEPT OF INTEGERS NATURAL NUMBERS LESS THAN AND GREATER THAN INTEGERS ABSOLUTE VALUE OPPOSITE OF AN INTEGER INTEGER ADDITION PRACTICAL EXAMPLE OF INTEGER ADDITION INTEGER SUBTRACTION INTEGER MULTIPLICATION PRACTICAL EXAMPLE OF INTEGER MULTIPLICATION CHALLENGE: INTEGER MULTIPLICATION CHALLENGE: INTEGER DIVISION INTEGER PROPERTIES HOW DO OPERATIONS WORK? APPLICATIONS OF INTEGERS DIVISIBILITY DIVISIBILITY TESTS STEP-BY-STEP EXAMPLES DIVISION ALGORITHM, EUCLIDEAN ALGORITHM THE DIVISION ALGORITHM DIVISION BY REPEATED SUBTRACTION INTEGER DIVISION (UNSIGNED) WITH REMAINDER NON-RESTORING DIVISION LARGE-INTEGER METHODS DIVISION BY A CONSTANT FUNDAMENTAL THEOREM OF ARITHMETIC APPLICATIONS GENERALIZATIONS CHAPTER 5 BINARY OPERATION DEFINITION OF BINARY OPERATIONS N-ARY OPERATION TABLES OF OPERATION PROPERTIES OF BINARY OPERATIONS MODULAR OPERATIONS MODULAR ARITHMETIC AS REMAINDERS OPERATIONS ON MATRICES MATRIX NOTATION MATRIX EQUALITY TYPES OF MATRICES MATRIX OPERATIONS OPERATIONS OF COMPLEX NUMBER EQUALITY OF TWO COMPLEX NUMBERS ADDITION OF COMPLEX NUMBERS SUBTRACTION OF THE COMPLEX NUMBERS MULTIPLICATION OF TWO COMPLEX NUMBERS EXAMPLE DIVISION OF TWO COMPLEX NUMBERS CHAPTER 6 INTRODUCTION TO MATHEMATICAL SYSTEMS SEMIGROUP SUBMONOIDS OF GROUPS ELEMENTARY SEMIGROUP THEORY GENERATORS AND SUBSEMIGROUPS GROUP BASICS OF GROUP GROUP HOMOMORPHISMS SYMMETRY OPERATIONS POINT GROUPS DIHEDRAL GROUPS LINEAR GROUPS CUBIC GROUPS PROPERTIES OF GROUPS FINITE GROUPS SUBGROUPS RING AND FIELD IDEALS AND FACTOR RINGS INTEGRAL DOMAINS ESSENTIAL CONCEPT OF FIELDS EXTENSION FIELDS SPLITTING FIELDS GEOMETRIC CONSTRUCTION