91Ó°ÊÓ

a) Verify that \(\mathbb{Z}\) and \(\mathbb{Q}\) are inductive sets. b) Give examples of inductive sets different from \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}\), and \(\mathbb{R}\).

Show that a) from a system of closed intervals covering a closed interval it is not always possible to choose a finite subsystem covering the interval; b) from a system of open intervals covering an open interval it is not always possible to choose a finite subsystem covering the interval; c) from a system of closed intervals covering an open interval it is not always possible to choose a finite subsystem covering the interval.

Show that a) every infinite set contains a countable subset; b) the set of even integers has the same cardinality as the set of all natural numbers; c) the union of an infinite set and an at most countable set has the same cardinality as the original infinite set; d) the set of irrational numbers has the cardinality of the continuum; e) the set of transcendental numbers has the cardinality of the continuum.

Show that a) the set of increasing sequences of natural numbers \(\left\\{n_{1}

Show that a) the set \(\mathcal{P}(X)\) of subsets of a set \(X\) has the same cardinality as the set of all functions on \(X\) with values 0,1 , that is, the set of mappings \(f: X \rightarrow\\{0,1\\}\); b) for a finite set \(X\) of \(n\) elements, card \(\mathcal{P}(X)=2^{n}\); c) taking account of the results of Exercises \(4 \mathrm{~b}\) ) and \(5 \mathrm{a}\) ), one can write \(\operatorname{card} \mathcal{P}(X)=2^{\text {card } X}\), and, in particular, card \(\mathcal{P}(\mathbb{N})=2^{\text {card } \mathbb{N}}=\operatorname{card} \mathbb{R}\); d) for any set \(X\) \(\operatorname{card} X<2^{\text {card } X}\), in particular, \(\quad n<2^{n} \quad\) for any \(n \in \mathbb{N}\)

Verify that the rational numbers \(\mathbb{Q}\) satisfy all the axioms for real numbers except the axiom of completeness.

a) Show that along with the unique representation of an integer as $$ \left(\alpha_{n} \alpha_{n-1} \ldots \alpha_{0}\right)_{3} $$ where \(\alpha_{i} \in\\{0,1,2\\}\), it can also be written as $$ \left(\beta_{n} \beta_{n-1} \ldots \beta_{0}\right)_{3} $$ where \(\beta \in\\{-1,0,1\\}\). b) What is the largest number of coins from which one can detect a counterfeit in three weighings with a pan balance, if it is known in advance only that the counterfeit coin differs in weight from the other coins?