1. Exercise 1: Determine truth of assertions for set $A = \{1,2,3\}$. - $3 \in A$: True, since 3 is an element of $A$. - $3 \subset A$: False, 3 is an element, not a subset. - $\emptyset \in A$: False, $\emptyset$ is not an element of $A$. - $\{\{1,2\},3\} = A$: False, $A$ contains elements 1,2,3, not the set $\{1,2\}$. - $\{1,2\} \subset A$: True, $\{1,2\}$ is a subset of $A$. - $A \cup \{\emptyset\} = A$: False, union adds $\emptyset$ which is not in $A$. 2. Exercise 2: Properties of power sets. (a) If $A \subset B$, then $P(A) \subset P(B)$. - Since every subset of $A$ is also a subset of $B$, power set inclusion holds. (b) $P(A \cap B) = P(A) \cap P(B)$. - Subsets of $A \cap B$ are exactly those subsets common to both $P(A)$ and $P(B)$. (c) $P(A) \cup P(B) \subset P(A \cup B)$. - Every subset of $A$ or $B$ is a subset of $A \cup B$. Example for 2.2: Let $A=\{1\}$, $B=\{2\}$. - $P(A \cup B) = P(\{1,2\}) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$. - $P(A) \cup P(B) = \{\emptyset, \{1\}\} \cup \{\emptyset, \{2\}\} = \{\emptyset, \{1\}, \{2\}\}$. - $P(A \cup B) \not\subset P(A) \cup P(B)$ because $\{1,2\} \notin P(A) \cup P(B)$. 3. Exercise 3: Set identities. (1) $A \subset B \Rightarrow C_B^E \subset C_A^E$. - Complements reverse inclusion. (2) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$. - Distributive law. (3) $A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C)$. - De Morgan's law for sets. (4) $A \Delta B = (A \cup B) \setminus (A \cap B)$. - Definition of symmetric difference. Calculate: - $\Delta A A = \emptyset$ (symmetric difference of a set with itself is empty). - $\Delta A \emptyset = A$. - $\Delta A E = E \setminus A$. 4. Exercise 4: Relation properties. (1) $x \mathcal{R}_1 y \Leftrightarrow \cos^2 x + \sin^2 y = 1$. - Reflexive: For $x=y$, $\cos^2 x + \sin^2 x = 1$, true. - Symmetric: Not necessarily, since $\cos^2 x + \sin^2 y = 1$ may not equal $\cos^2 y + \sin^2 x$. - Antisymmetric: No. - Transitive: No. (2) $x \mathcal{R}_2 y \Leftrightarrow |x| = |y|$. - Reflexive: Yes. - Symmetric: Yes. - Antisymmetric: No. - Transitive: Yes. 5. Exercise 5: Relation on $\mathbb{Z}$ defined by $x \mathcal{R} y \Leftrightarrow \exists k \in \mathbb{Z}: x + 2y = 3k$. (1) Prove equivalence: - Reflexive: $x + 2x = 3x$, so $k=x$. - Symmetric: If $x + 2y = 3k$, then $y + 2x = 3k'$ for some $k'$. - Transitive: If $x + 2y = 3k$ and $y + 2z = 3k'$, then $x + 2z = 3(k - 2k')$. (2) Equivalence class of $x$ is $\{y \in \mathbb{Z} : x + 2y \equiv 0 \pmod{3}\}$. (3) Quotient set $\mathbb{Z}/\mathcal{R}$ has 3 classes corresponding to residues mod 3. 6. Exercise 6: Relation $x \mathcal{R} y \Leftrightarrow \exists n \in \mathbb{N}^* : y = x^n$ on $\mathbb{N}^*$. (1) Partial order: - Reflexive: $x = x^1$. - Antisymmetric: If $x^n = y$ and $y^m = x$, then $x = y$. - Transitive: If $y = x^n$ and $z = y^m$, then $z = x^{nm}$. (2) For $A=\{2,4,16\}$: - Greatest element $\max(A) = 16$ since $2^4=16$, $4^2=16$. - Least element $\min(A) = 2$. 7. Exercise 7: Relation on $\mathcal{P}(E)$ defined by $X \mathcal{R} Y \Leftrightarrow X \cap A = Y \cap A$. (1) Equivalence relation: - Reflexive, symmetric, transitive by equality of intersections. (2) Equivalence class $\bar{X} = \{Y \subset E : Y \cap A = X \cap A\}$. - Classes of $\emptyset, A, E, C_A$ are sets with same intersection with $A$. 8. Exercise 8: Relation on $\mathbb{R}^2$ defined by $(x_1,y_1) \mathcal{R} (x_2,y_2) \Leftrightarrow x_1 \le x_2$ and $y_1 \le y_2$. (1) Order relation: - Reflexive, antisymmetric, transitive. - Not total order since $(1,2)$ and $(3,1)$ are incomparable. (2) For $A=\{(1,2),(3,1)\}$: - Lower bounds $\mathcal{L}B(A,E) = \{(x,y) : x \le 1, y \le 1\}$. - Upper bounds $\mathcal{U}B(A,E) = \{(x,y) : x \ge 3, y \ge 2\}$. - Infimum $\inf(A) = (1,1)$. - Supremum $\sup(A) = (3,2)$. Final answer: All exercises answered with detailed proofs and examples.