Schaum’s outline of theory and problems of set theory and related topics

Seymour Lipschutz0070381593, 9780070381599

If you want top grades and thorough understanding of set theory and related topics, this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you 530 accompanying related problems with fully worked solutions. You also get plenty of practice problems to do on your own, working at your own speed. (Answers at the back show you how you’re doing.) This new edition features improved problems in the ordinals, cardinals, and transfinite series chapters, plus new coverage of real numbers and integers.

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Table of contents :
Front cover……Page 1
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Contents……Page 7
1.2 Sets and elements……Page 9
1.4 Subsets……Page 11
1.5 Venn diagrams……Page 12
1.6 Set operations……Page 13
1.7 Algebra of sets, duality……Page 15
1.8 Finite sets, counting principles……Page 16
1.9 Classes of sets, power sets……Page 19
1.10 Arguments and Venn diagrams……Page 20
1.12 Axiomatic development of set theory……Page 21
2.2 Real number system $mathbb{R}$……Page 42
2.3 Order and inequalities……Page 43
2.4 Absolute value, distance……Page 44
2.5 Intervals……Page 45
2.6 Bounded sets, completion property……Page 47
2.7 Integers $mathbb{Z}$ (optional material)……Page 48
2.8 Greatest common divisor, euclidean algorithm……Page 50
2.9 Fundamental theorem of arithmetic……Page 52
3.2 Product sets……Page 72
3.3 Relations……Page 73
3.4 Pictorial representations of relations……Page 75
3.5 Composition of relations……Page 76
3.6 Types of relations……Page 78
3.7 Closure properties……Page 79
3.9 Equivalence relations……Page 81
3.10 Partial ordering relations……Page 83
3.11 $n$-ary relations……Page 84
4.2 Functions……Page 102
4.3 Composition of functions……Page 104
4.4 One-to-one, onto, and invertible functions……Page 106
4.5 Mathematical functions, exponential and logarithmic functions……Page 108
4.6 Recursively defined functions……Page 111
5.3 Indexed collections of sets……Page 125
5.4 Sequences, summation symbol……Page 127
5.5 Fundamental products……Page 128
5.6 Functions and diagrams……Page 129
5.7 Special kinds of functions, fundamental factorization……Page 130
5.8 Associated set functions……Page 132
5.9 Choice functions……Page 133
5.10 Algorithms and functions……Page 134
5.11 Complexity of algorithms……Page 135
6.2 One-to-one correspondence, equipotent sets……Page 149
6.3 Denumerable and count able sets……Page 151
6.4 Real numbers $mathbb{R}$ and the power of the continuum……Page 153
6.5 Cardinal numbers……Page 154
6.6 Ordering of cardinal numbers……Page 155
6.7 Cardinal arithmetic……Page 158
7.2 Ordered sets……Page 174
7.3 Set constructions and order……Page 176
7.4 Partially ordered sets and Hasse diagrams……Page 178
7.5 Minimal and maximal elements, first and last elements……Page 179
7.7 Supremum and infimum……Page 180
7.8 Isomorphic (similar) ordered sets……Page 183
7.9 Order types of linearly ordered sets……Page 184
7.10 Lattices……Page 185
7.11 Bounded, distributive, complemented lattices……Page 186
8.2 Well-ordered sets……Page 212
8.4 Limit elements……Page 213
8.5 Initial segments……Page 214
8.7 Comparison of well-ordered sets……Page 215
8.9 Inequalities and ordinal numbers……Page 216
8.10 Ordinal addition……Page 217
8.11 Ordinal multiplication……Page 219
8.13 Auxiliary construction of ordinal numbers……Page 220
9.2 Cartesian products and choice functions……Page 227
9.5 Cardinal and ordinal numbers……Page 228
9.7 Paradoxes in set theory……Page 229
10.2 Propositions and compound propositions……Page 237
10.3 Basic logical operations……Page 238
10.4 Propositions and truth tables……Page 240
10.5 Tautologies and contradictions……Page 241
10.7 Algebra of propositions……Page 242
10.8 Conditional and biconditional statements……Page 243
10.9 Arguments……Page 244
10.11 Propositional functions, quantifiers……Page 246
10.12 Negation of quantified statements……Page 249
11.2 Basic definitions……Page 260
11.3 Duality……Page 261
11.5 Boolean algebras as lattices……Page 262
11.6 Representation theorem……Page 263
11.8 Sum-of-products form for boolean algebras……Page 264
11.9 Minimal boolean expressions, prime implicants……Page 267
11.10 Karnaugh maps……Page 269
INDEX……Page 289
Back cover……Page 291