Abstract
An important concept in mathematics and computer science is that of a collection of objects, or a set. In this chapter, typed set theory is introduced and standard set operators such as union, intersection, set complement and set difference are formally defined. Various properties about sets and the set operators are also proved using predicate logic. The chapter closes with a discussion of the importance of types in set theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A stronger condition is to ensure that, if the formula is reduced to use only negation, conjunction and disjunction, then there is an even number of negations above α for it to be safe to replace it by β. For example, if α⇒β, then α∧¬(P∨¬α) would imply β∧¬(P∨¬β), since all instances of α appear under an even number of negations (zero and two, respectively).
- 2.
Each of the n elements of the set may either be placed or not in a subset. This gives two possibilities for each element, and thus 2×2×⋯×2 (one for each element) possibilities in total. Since we have n elements, we get 2n.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pace, G.J. (2012). Sets. In: Mathematics of Discrete Structures for Computer Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29840-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-29840-0_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29839-4
Online ISBN: 978-3-642-29840-0
eBook Packages: Computer ScienceComputer Science (R0)