Page 397 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Index 383
pairwise disjoint, 153, 214 recursive
partial order, 190, 254, 267, 268–269, definition, 280
314 procedure, 274
strict, 211 refine, 225
partition, 214, 215 reflexive, 184
Pascal, Blaise, 288 reflexive closure, 202
Pascal’s triangle, 288 reflexive symmetric closure, 212
perfect number, 5 relation
Pigeonhole Principle, 313 antisymmetric, 212
polynomial, 299 asymmetric, 210, 346
power set, 75–76, 119, 318 binary, 182, 242
premise, 8 composition of, 176, 180–181
preorder, 225 domain of, 173
prime number, 1, 74, 94, 156–158 identity, 183, 203–204, 213
largest known, 5 inverse of, 173
Mersenne, 5 irreflexive, 204
twin, 6 range of, 311
proof, 1, 84 reflexive, 184
by cases, 137 symmetric, 184
by contradiction, 96, 97, 98, 99 transitive, 184
Proof Designer, 102, 373–374 remainder, 290, 139–140
Proof Designer, Preface and Appendix, restriction, 234, 243
373–374 rule of inference, 107
proof strategy Russell, Bertrand, 83
for a given of the form Russell’s Paradox, 83
¬P, 99–100, 102
P ∧ Q, 124 Schr¨oder, Ernst, 322
P ∨ Q, 142–143 sequence, 297–298
P → Q, 102 set, 27. See also countable set; denumerable
P ↔ Q, 126 set; empty set (or null set); family of
∀xP(x), 115 sets; finite set; index set; infinite sets;
∃xP(x), 115 power set; subset; truth set
∃!xP(x), 152 -notation, 281–282
for a goal of the form smallest element, 184, 191
¬P, 95, 96 strict partial order, 204
P ∧ Q, 124 strict total order, 204
P ∨ Q, 138, 141–142 strictly dominates, 322
P → Q, 88, 90, 91 strong induction, 288–295
P ↔ Q, 126 subset, 40, 184
∀xP(x), proper, 203
∀n ∈ NP(n), 260, 289 sufficient condition, 50
∃xP(x), surjection, 236
∃!xP(x), 149, 150 symmetric closure, 204, 205
proper subset, 203 symmetric difference, 36, 43, 154
symmetric transitive closure, 40, 212, 184
quantifier, 55–64
bounded, 68, 57 tautology, 23
existential, 58 laws, 23
negation laws, 65, 66, 69 theorem, 85
unique existential, 146 total order, 190, 269–270, 275
universal, 55 strict, 204
quotient, 290 transitive,
transitive closure, 204, 209, 300
range, 173, 239 trichotomy, 204
rational number, 31, 311, 326, truth set, 27, 30, 37, 163
328 truth table, 14–23
real number, 31, 297 truth value, 14