We begin BookofProofs with a branch of mathematics dedicated to the set theory. 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 one country to the “people” of this country, we distinguish between “our belongings” and the “belongings of others” or we just collect (or buy) things we call “fruits” and put them into a basket on our table. In contrast to things or persons that we can count (like citizens, belongings, fruits, etc.), there are also things that we cannot count, for instance, we might measure “how much” air, water, sugar, etc. we can put into a container. Once we put them into this container, say a bottle, we are also able to easily distinguish between the air inside and outside the bottle. Whenever we can distinguish objects, count how many they are (or measure how much we have out of them), we intuitively use a concept, known in mathematics as a set.
Today’s mathematics is characterized by sets and their language. For instance, the concepts of numbers, including natural numbers, integers, rational numbers, and real numbers can be traced back to sets. Basic concepts of almost every mathematical discipline can be defined using sets, for instance, the topological space in topology, the algebraic structures in algebra, and many more.
Set theory is the systematic study of principles governing sets, which are the most basic mathematical objects. The four basics even a non-mathematician has to know about sets are:
- Sets can be completely described (defined) by either listing all their elements or defining the properties, the elements must fulfill.
- An object belongs to a set, if it is listed in the set or if it fulfills the defined properties.
- Sets do not allow duplicates (taking an element out of the set removes this element from the set completely).
- If you remove every element from the set, it becomes empty, (but it still exists, like it was an “empty container”).
Theoretical minimum (in a nutshell)
BookofProofs starts with set theory. It needs, however, 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 later 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 $x$”.
- equality $\phi=\psi$ and inequality $\phi\neq\psi.$
- 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!
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