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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.
For the following exercises to solve, you actually only need to know a basic understand what means a set to be countable resp. uncountable. As a short reminder, a set is countable if there is a bijection between the same and the set of natural numbers. It is uncountable in case only a surjection but no injection exists onto the natural numbers. More details you may find here. To underline the importance of these definitions let me mention that the set of rational numbers is countable and the set of irrationals is uncountable. This probably most intensively shows the relevance of it.
As always, I suggest to try to solve the following problems first on your own. Even if you do not come up with a solution, this way you will have a maximal learning effect!
So, the solution uses the fact that the countable union of finite sets again is countable. This is a theorem typically provided in any lecture about basic set theory. A proof of an even more general result can be seen here.
The solution relies on the famous fact that the real numbers are uncountable. A statement which proof you may find here.
Thanks for reading!