Abstract
This chapter treats two interrelated topics in discrete mathematics: elementary Set Theory (set notation; subsets, partitions, and power sets; and unions, intersections, set differences, and Cartesian products) and the theory of Combinatorics (sequential counting, permutations, and combinations). Strong connections to Propositional Logic (Chapter 1) are demonstrated, and applications are made to binomial expansion, discrete probability, and everyday counting. The material on Set Theory also provides the theoretical foundation for topics later in the text on infinite sets (Chapter 5), functions (Chapter 6), and relations (Chapters 6 and 7).
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Notes
- 1.
In Section 5.3 we’ll see why we should restrict sets to those that can be formed inside already existing sets.
- 2.
Venn attributes circle diagrams to Euler. Euler may have gotten the idea from his teacher, Jean Bernoulli, who in turn may have been indebted to his collaborator Leibniz (the linkage isn’t clear). Leibniz used them to exhibit logical relations among various classes. Versions prior to Venn, though, were less general and less versatile than his.
- 3.
There is more here than meets the eye, because we already have a fixed meaning for equals (see Section 2.3). The forward part of this definition follows from FOL’s inference rules for identity and so can be proved. The backward part, however, must be asserted as an axiom (see Section 5.3). Taken together, though, they specify how equality is used in Set Theory, so we’ll treat it here as a definition of set equality.
- 4.
Note that this and later propositions are intended as universal statements, though we’ve omitted the quantifiers \(\forall S\,\forall T\) in the interest of readability.
- 5.
This assumes no set has itself as an element. We’ll touch on this briefly in Section 5.3.
- 6.
There are mathematical entities, called multisets or bags, however, in which multiplicity is taken into account. We’ll make use of this idea in Section 4.3.
- 7.
This is named after the seventeenth-century French mathematician Blaise Pascal, who investigated its properties. It was known several centuries earlier, however, both to Arabic mathematicians and Chinese mathematicians.
- 8.
This Stars-and-Bars Method for counting unordered collections with repetition was popularized by William Feller in his classic 1950 treatise on probability.
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Jongsma, C. (2019). Basic Set Theory and Combinatorics. In: Introduction to Discrete Mathematics via Logic and Proof. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-25358-5_4
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DOI: https://doi.org/10.1007/978-3-030-25358-5_4
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