EQUIVALENCE RELATION DEFINED NATURAL GROUPING ELEMENTS RELATED FORMING EQUIVALENCE CLASSES EQUIVALENCE CLASS ELEMENT SUBSET EQUIVALENT IT FOLLOWS THE DEFINITION OF EQUIVALENCE RELATIONS PARTITION THE SET SOMETIMES QUOTIENT QUOTIENT SPACE DENOTED CONNECTION SPACES LINEAR ALGEBRA TOPOLOGY QUOTIENT GROUP HOMOGENEOUS SPACE QUOTIENT RING QUOTIENT MONOID CATEGORIES BINARY RELATION PROPERTIES REFLEXIVITY SYMMETRY TRANSITIVITY ALTERNATIVE NOTATION BE USED TO DENOTE WITH RESPECT TO SAID TO CLASS MODULO SURJECTIVE MAP ONTO MAPS CANONICAL SURJECTION CANONICAL PROJECTION MAP CHOSEN DEFINES INJECTIVE MAP SECTION FOR EVERY REPRESENTATIVE OF CHOOSING THE OTHER ONES MODULAR ARITHMETIC CONSIDER INTEGERS BY IF MULTIPLE INTEGER UNIQUE SMALLER THAN THE CLASS REPRESENTATIVE MORE OR LESS FACT THE REMAINDER THE DIVISION SAME COLOR PARTICULAR GREEN NATURALLY IDENTIFIED CAR COLORS CARDINALITY NUMBER RECTANGLES AREA FOR EACH POSITIVE REAL NUMBER IF AND ONLY IF DIFFERENCE EVEN NUMBER RELATION RISE CONSISTING OF EVEN NUMBERS THE OTHER ODD NUMBERS PAIRS DEFINE PAIR RATIONAL NUMBER GIVE FORMAL DEFINITION OF RATIONAL NUMBERS CONSTRUCTION GENERALIZED THE FIELD FRACTIONS INTEGRAL DOMAIN THE LINES EUCLIDEAN PLANE MEANS PARALLEL LINES LINES PARALLEL TO EACH ITSELF DETERMINES POINT AT INFINITY MEMBER EQUAL DISJOINT BELONGS TO ONE AND ONLY ONE CONVERSELY COMES BELONG TO IN OTHER WORDS STATEMENTS DIRECTED GRAPH SYMMETRIC UNDIRECTED GRAPH VERTICES GRAPH JOIN MAXIMAL CONNECTED COMPONENTS PROPERTY OF SUCH THAT TRUE THE PROPERTY INVARIANT CASE FUNCTION MORPHISM CLASS INVARIANT SIMPLY CHARACTER THEORY FINITE GROUPS AUTHORS INVERSE IMAGE THE KERNEL ARGUMENTS VALUES TOPOLOGICAL SPACE THE ORIGINAL CREATE ABSTRACT ALGEBRA CONGRUENCE RELATION UNDERLYING SET ALGEBRA INDUCE VECTOR SPACE TAKING HOMOMORPHISM LINEAR MAP TERM QUOTIENT MODULE THE MORE GENERAL CASES ANALOGY ORBITS GROUP ACTION THE ACTION PARTICULARLY THE GROUP ACTION THE RIGHT COSET SUBGROUP ARISE LEFT THE LEFT COSETS RIGHT NORMAL SUBGROUP TOPOLOGICAL GROUP ACTING TRANSLATION THE SENSES GROUP ACTIONS SIMULTANEOUSLY FOR ANY INTENT GENERALLY COMPARE TYPE INDUCES STRUCTURE KIND SENSE THE STUDY LEAD EQUIVALENCE PARTITIONING METHOD SOFTWARE TESTING THE POSSIBLE PROGRAM BEHAVIOR THE PROGRAM LIE GROUP TRANSVERSAL MATERIAL TEXT DEALING THE FUNDAMENTALS OF PROOF