Gentle Introduction to the Art of Mathematics

cover_thumbnail.pdf......Page 1
GIAM 2.0.pdf......Page 2
Basic sets......Page 24
Definitions: Prime numbers......Page 35
More scary notation......Page 44
Even and odd......Page 47
Decimal and base-n notation......Page 48
Divisibility......Page 50
Floor and ceiling......Page 51
Div and mod......Page 52
Binomial coefficients......Page 53
Some algorithms......Page 61
Rational and irrational numbers......Page 71
Relations......Page 76
Predicates and Logical Connectives......Page 82
Implication......Page 96
Logical equivalences......Page 102
Two-column proofs......Page 115
Quantified statements......Page 119
Deductive reasoning and Argument forms......Page 128
Validity of arguments and common errors......Page 137
Direct proofs of universal statements......Page 146
More direct proofs......Page 159
Contradiction and contraposition......Page 163
Disproofs......Page 169
By cases and By exhaustion......Page 174
Existential statements......Page 183
Basic notions of set theory......Page 192
Containment......Page 199
Set operations......Page 204
Venn diagrams......Page 214
Russell's Paradox......Page 223
The principle of mathematical induction......Page 228
Formulas for sums and products......Page 238
Other proofs using PMI......Page 249
The strong form of mathematical induction......Page 257
Relations......Page 260
Properties of relations......Page 270
Equivalence relations......Page 278
Ordering relations......Page 287
Functions......Page 297
Special functions......Page 311
Counting......Page 322
Parity and Counting arguments......Page 339
The pigeonhole principle......Page 354
The algebra of combinations......Page 360
Equivalent sets......Page 372
Examples of set equivalence......Page 378
Cantor's theorem......Page 390
Dominance......Page 399
CH and GCH......Page 408
Proof techniques IV — Magic......Page 414
Morley's miracle......Page 416
Five steps into the void......Page 424
Monge's circle theorem......Page 436
References......Page 442
GNU Free Documentation License......Page 444
Index......Page 446