Basic Set Theory by Exercises: Products and Unions.

2 min readMay 19, 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.

The following question can be categorized as being easy since it requires only most basic knowledge of set theory itself. Though, it may appear a little tricky to directly find a good approach to the solution.


Show the following statement to be true:

This roughly states that the above union operator translates into an ‘OR’ on the level of factors.

Remember, the Cartesian product of two sets A and B is defined as the set of all tuples (a, b) with a being an element of A and b being an element of B.

The union of two sets A and B is defined to be the set that exactly contains all elements of A and B.

Good explanations you may also find here: (product, union).

Note, both of the above definitions are so called ‘naive’ definitions. One also can give more precise definitions based on the axioms of set theory, but for the exercise the naive view suffices entirely.


Before reading the proof I suggest trying it yourself. Even if you do not come to any solution, you will have a better learning experience when having had some own initial thoughts on the problem.

(if the link is not displayed, you can go here)

If you are new to mathematical proofing, then let me mention that the above uses the so called ‘proof by contradiction’.

If you find an easier solution, you are welcome to tell us in the comments.

Thanks for reading!




I am a Software Developer - Rust, Java, Python, TypeScript, SQL - with strong interest doing research in pure and applied Mathematics.