Sets and Data Structures
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The second chapter goes into the basics of set theory and data structures. The first section takes an excursion into propositions and logic, including the definition and application of truth tables. Tautologies are defined and the basic laws of logic are introduced.
The second section treats the basic ideas of set theory: finite, infinite, and empty sets, the operations of union, intersection, and relative complements, and disjointness are defined, and the properties of the operations are explored. Then proof methods in set theory are studied, including the use of truth tables and Venn diagrams. There is a discussion of syllogisms using Venn diagrams. In Section 2.4 further set operations, such as the Cartesian product and relative difference, are introduced and their properties are studied.
The chapter concludes with a very important section on mathematical induction. After the introduction of the principle of mathematical induction, there is a discussion of the related well-ordering principle. There are a number of examples of the use of induction in the proof of mathematical results. The Fibonacci numbers are introduced, and the flaws in the purported “proofs” by induction are explored.