282 Index Limit definition, 172, 177 existence, 167 of a function, 170, 174 nonexistence, 178 of a sequence, 159, 164­165, 167­169, 177 Logic, 1 Logical equivalence, 16 Function (Cont.) range of a, 89 simple, 180 surjective. See Onto function G Gauss' proof, 15 Greatest common divisor, viii Group Abelian, 134, 139­140, 142, 154, 157­158 commutative, 132, 155 isomorphic, 152­153, 155­156, 158 Mod M and Arithmetic, 142­148 permutation, 156 size and structure, 138­142 symmetric, 148­151, 150 theory, 138 Groups, 118­159 M Mathematical induction, 42­51 Matrices, 119, 125, 157 Memorization, 1 Mod M and Arithmetic groups, 142­148 Modus ponens, 8 Monoid, 132, 146, 150 Monotone sequence, 159, 161 Multiple conclusions, 61­66 Multiple hypotheses, 58­61 Multiplication table, 142­143, 143­144 H Hypothesis definition, 2 implicit, 10­11, 64 inductive, 43­44, 45 multiple, 58­61 understanding, 3 N Negation of a statement, 16­21 Negative sequence, 159 O Odd function, ix One-to-one function, 90­91, 90­91, 94­95, 180 Onto function, 90­91, 90­92, 180, 182 Operation table, 120, 122, 124, 137, 139, 146, 150 Operations, properties of, 80 Order relation, 106 I Identity element, 128, 130, 135, 139­141, 153 function, 129, 149 for the operation, 128 If and only if theorems, 33­40 If/then statements, 12 Image of an element, 99 of an interval, 98­99 of a set, 86 Implicit hypothesis, 10­11, 64 Increasing function, viii Increasing sequence, 159­160, 162 Indexed families of sets, 84­87 Indexing set, 84 Induction, mathematical, 42­51 Inequalities, 49­50, 73, 166, 173, 175 Injective function. See One-to-one function Intersection of sets, 74, 76, 85 Inverse of an element, 128, 130, 135, 137, 154, 157 of a function, 55­56, 97, 154 of a statement, 21 Isomorphic groups, 152­153, 155­156, 158 Isomorphism, 152­156, 158 P Paradox, 2 Partition, 115­116 Permutation group, 156 Permutations, 148­152, 158 Polynomial function, 53 Positive sequence, 159 Pre-image of a set, 100 Prime number, 29, 56­57, 148 Principle of Mathematical Induction definition, 43 Strong, 46­47 Weak, 46 Proof construction, 13, 19, 43, 171 by contradiction, 25­32, 52 by contrapositive, 21­25 definition, 2 design, 4 direct, 9 Gauss', 15 by induction, 12, 42­51 sound, 5 Properties of operations between sets, 79 L Least common multiple, vii Lemma, 3