It seems to be a basic ability of the human mind to combine given objects into a whole. For instance, we combine the citizens of a country to its “people”, we distinguish between “cars”, “animals”, “fruits” and so on. These entities are examples of an abstract concept mathematicians call a “set”. Mathematical research, especially at the beginning of the 20th century, has shown that “sets” are fundamental for developing other mathematical theories. Nowadays, mathematics is permeated by sets and their language. For instance, numbers, algebraic structures, topological structures, functions, relations, to mention only a few further fundamental mathematical concepts, turn out not only to make use of sets in order to be explained but they are sets.
Set theory is the mathematical discipline specifying precisely what a “set” is and studies systematically the principles, which govern sets. It also provides a standard notation and many basic mathematical results, which are re-used in all other mathematical disciplines, including relations and functions. Moreover, set theoretical-research has led to many fascinating insights regarding infinite sets and their handling belongs to the standard repertoire in today’s mathematics.
Theoretical minimum (in a nutshell)
The four basics even a non-mathematician has to know about sets are:
- Sets are completely described (defined) by either listing all their elements or defining the properties, the elements must fulfill.
- An object is an element of a set if it is listed in its definition or fulfills the set’s defined properties. This has important consequences, for instance, a set in which elements are duplicated and another set in which the same elements are not duplicated are equal.
- A set can be empty, (like an “empty container”) and there is only one empty set (i.e. an empty container of fruits and an empty container of cars are essentially the same).
- A set can have a finite or infinite number of elements.
We will need some formal notation to precisely describe the objects related to sets. The notations we will be using are a kind of mathematical metalanguage which can itself be formally defined. This will be done in the branch Logic. At this stage, the reader should be familiar with the following concepts and notations of propositional logic:
- notations for variables and propositions, $\phi, \psi,\ldots, x,y,\ldots ,$
- negation of a proposition $\neg \phi,$
- disjunction $\phi\vee \psi$ and conjunction $\phi\wedge \psi$ of propositions,
- implication $\phi\Rightarrow \psi$ and equivalence $\phi\Leftrightarrow\psi$ of propositions,
- existential quantifier for variables $\exists x$ : read “there exists an $x$”,
- universal quantifier for variables $\forall x$ : read “for all $x$”.
- Please refer also to examples for the use of quantifiers.
Concepts you will learn in this part of BookofProofs
- What are the Zermelo-Fraenkel Axioms and how to define sets using them?
- Which are the basic set operations and how to “calculate” with sets?
- What are relations, and in particular equivalence relations?
- What are functions?
- How can sets be “compared with each other” and start a fascinating journey into infinite sets using ordinals?
- How can sets be used to “count things” and learn even more astonishing things about infinite sets, called cardinals?
One final remark: The study of set theory is highly intuitive and accessible to the undergraduates. However, understanding of the meaning of Zermelo-Fraenkel Axioms, ordinals and cardinals might be somewhat demanding at the beginning. It’s definitely worth the effort!
| | | | created: 2014-02-01 19:09:07 | modified: 2018-03-22 22:46:25 | by: bookofproofs