Basic Set Theory by Exercises: First Order Logic — Propositional Logic

5 min readJun 4, 2022

Set theory is a very fundamental and interesting part of mathematics. Typically its basics are being taught in the first semester of a course in mathematics. Everybody who uses mathematics in some form should have a basic knowledge of this subject. In this series of articles I want to provide some nice exercises together with their solutions, that I discovered at various places. It is like everywhere, learning by doing makes you a master, but not holding some certificate or title.

Mathematical logic can be seen has the common base of everything in mathematics, though this fact might not be so immanent when doing subjects like analysis or probability theory. With set theory this is different. In order to understand the basics of this subject it is absolutely necessary to also understand the basics of first order logic. The latter link provides a good overview and introduction. As usually, here we will focus on solving some interesting problems to manifest the learned theory.

Wrap-up of logic:

Most probably you have already seen formulas of the sort like this:

And certainly you already know the common interpretation of the appearing symbols.

When starting to learn logic it is most advisable to forget these common interpretations for a moment. The reason is, these interpretations do not belong to the subject ‘logic’. Logic itself deals with the formalization to deal with sets of formulas like the one above. So it views the above formula as a sequence of symbols, though distinguishing between certain types of the latter.

For instance A, B, C are called variables. This is just a name for a type of symbol (do not think of variables like you are used to, for a moment). Further, the symbols

are of a type that is called propositional connectives.

And the

are so called quantifiers.