Lesson 2

Advanced Problems.

1. (20 points) Is it true that the set of all equivalence relations on the set of natural numbers, in which each equivalence class is finite, has cardinality equal to the continuum?
2. (20 points) Prove that the function \(f: \mathbb{Z}\to \mathbb{Z}\), defined by the rules \[f(2k) = 0, \quad f(2k + 1) = 1,\] cannot be represented as a sum of two bijections \(g, h \colon \mathbb{Z}\to \mathbb{Z}\).
3. (20 points) Let \(S\) be a collection of non-intersecting figures \(F \subseteq \mathbb{R}^{2}\) each of which consists of two circles (of positive radius) touching each other. Prove that \(S\) is either finite or countable.
4. (20 points) Prove that the set of continuous functions \(\mathbb{R}\to \mathbb{R}\) has the cardinality of the continuum.
5. (20 points) Let \(\mathcal{F}\) be the family of all closed subsets of \([0,1]\). Show \(|\mathcal{F}|=|\mathbb{R}|\).
6. (20 points) Does there exist a family of subsets of \(\mathbb{Z}_{\ge 0}\) such that the intersection of any two distinct sets in this family is finite and the cardinality of the family is the continuum?
7. (20 points) An almost disjoint family of infinite subsets of \(\mathbb{N}\) is a family \(\mathcal{F}\subseteq \mathcal{P}(\mathbb{N})\) such that for any two distinct \(A, B \in \mathcal{F}\), the intersection \(A \cap B\) is finite. The family is called a maximal almost disjoint family (or MAD family) if it cannot be extended by any other infinite subset of \(\mathbb{N}\). Prove that every MAD family is uncountable.
8. (20 points) Does there exist a family of subsets of \(\mathbb{Z}_{\ge 0}\) such that for any two sets in this family, one is contained in the other and the cardinality of the family is continuum?
9. (20 points) The square \([0,1]\times[0,1]\) is represented as the union of two sets \(C\) and \(D\). Prove that either \(C\) or \(D\) has a cardinality of the continuum.
10. (20 points) Partition \(\mathbb{R}\) into two disjoint dense sets \(A,B\), each of size \(|\mathbb{R}|\).