Problem 1
Prove that if \(\mathrm{A} \cup \mathrm{B}=A\) and \(\mathrm{A} \mathrm{n} \mathrm{B}=\mathrm{A}\), then \(\mathrm{A}=\mathrm{B}\).
Problem 2
Show that in general \((\mathrm{A}-\mathrm{B}) \cup \mathrm{B} \neq \mathrm{A}\).
Problem 3
Let \(A=\\{2,4, \ldots, 2 n, \ldots)\) and \(\mathrm{B}=\\{3,6, \ldots, 3 n, \ldots)\). Find \(A \cap \mathrm{B}\) and \(\mathrm{A}-\mathrm{B}\).
Problem 4
Prove that a) \((\mathrm{A}-\mathrm{B}) \cap C=(\mathrm{A} \cap C)-(B \cap C)\) b) \(A \triangle B=(\mathrm{A} \cup B)-(\mathrm{A} \cap \mathrm{B})\).
Problem 6
Let \(A_{n}\) be the set of all positive integers divisible by \(\mathrm{n}\). Find a) \(\bigcup_{n=2}^{\infty} A_{n}\); b) \(\bigcap_{n=2}^{\infty} A_{n}\).
Problem 8
Let \(\mathrm{A}\), be the set of points lying on the curve $$ y=\frac{1}{x^{\alpha}} \quad(0<\mathrm{x}<\infty) $$ What is $$ \bigcap_{\alpha \geqslant 1} A, ? $$
Problem 9
Let \(\mathrm{y}=\mathrm{f}(\mathrm{x})=(\mathrm{x})\) for all real \(\mathrm{x}\), where \((\mathrm{x})\) is the fractional part of \(x\). Prove that every closed interval of length 1 has the same image under \(f\). What is this image? Is \(f\) one-to-one? What is the preimage of the interval \(\frac{1}{4} \leqslant \mathrm{y} \leqslant \frac{3}{4} ?\) Partition the real line into classes of points with the same image.
Problem 10
Given a set \(M\), let \(\mathscr{R}\) be the set of all ordered pairs on the form (a, a) with a \(\in \mathrm{M}\), and let \(a R b\) if and only if \((\mathrm{a}, \mathrm{b}) \in \mathscr{R}\). Interpret the relation \(\mathrm{R}\).
Problem 11
Give an example of a binary relation which is a) Reflexive and symmetric, but not transitive; b) Reflexive, but neither symmetric nor transitive; c) Symmetric, but neither reflexive nor transitive; d) Transitive, but neither reflexive nor symmetric.
