1. **Problem Statement:** Classify the given statements about set $A = \{1, 2, 3\}$ as true or false. 2. **Statements:** - $2 \in A$ means 2 is an element of $A$. - $3 \subset A$ means 3 is a subset of $A$ (which is incorrect since 3 is an element, not a set). - $\emptyset \in A$ means the empty set is an element of $A$. - $\{\emptyset\} \subset A$ means the set containing the empty set is a subset of $A$. - $A \cup \{\emptyset\} = A$ means the union of $A$ and the set containing the empty set equals $A$. 3. **Evaluations:** - $2 \in A$ is **true** because 2 is an element of $A$. - $3 \subset A$ is **false** because 3 is an element, not a subset. - $\emptyset \in A$ is **false** because the empty set is not an element of $A$. - $\{\emptyset\} \subset A$ is **false** because $\{\emptyset\}$ contains the empty set which is not in $A$. - $A \cup \{\emptyset\} = A$ is **false** because adding $\{\emptyset\}$ adds a new element, so the union is $\{1,2,3,\emptyset\}$. --- 4. **Problem Statement:** Given $B = \{1, 2\}$ and $C = \{1, 3\}$, calculate: (a) Set operations: - $B \cap C = \{1\}$ (elements common to both) - $B \cup C = \{1, 2, 3\}$ (all elements in either set) - $C^B$ is not standard notation; assuming complement of $B$ in $C$, $C \setminus B = \{3\}$ - $C_A$ and $AB$, $BAC$ are unclear notation; assuming $C_A$ means $C \cap A$, $AB$ means $A \cap B$, $BAC$ means $B \cap A \cap C$: - $C \cap A = \{1, 3\} \cap \{1, 2, 3\} = \{1, 3\}$ - $A \cap B = \{1, 2, 3\} \cap \{1, 2\} = \{1, 2\}$ - $B \cap A \cap C = \{1, 2\} \cap \{1, 2, 3\} \cap \{1, 3\} = \{1\}$ (b) Cartesian products and power sets: - $B \times C = \{(1,1), (1,3), (2,1), (2,3)\}$ - $B \times \emptyset = \emptyset$ (product with empty set is empty) - $B \times \{\emptyset\} = \{(1, \emptyset), (2, \emptyset)\}$ - $P(B) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$ - $P(P(B))$ is the power set of $P(B)$, which has $2^{4} = 16$ subsets. --- 5. **Problem Statement:** Let $R$ be a binary relation on $\mathbb{R}^3$ defined by $$ (x,y,z) R (a,b,c) \iff (x - a) \leq (b - y) \text{ and } z = c. $$ Show $R$ is an order relation and determine if it is total. 6. **Order relation properties:** - Reflexive: For all $(x,y,z)$, check if $(x,y,z) R (x,y,z)$. - $(x - x) = 0 \leq (y - y) = 0$ and $z = z$, so reflexive. - Antisymmetric: If $(x,y,z) R (a,b,c)$ and $(a,b,c) R (x,y,z)$, then $(x,y,z) = (a,b,c)$. - From $z = c$ and $c = z$, $z = c$. - From inequalities $(x - a) \leq (b - y)$ and $(a - x) \leq (y - b)$, adding gives $0 \leq 0$, so equality holds. - Hence, $x = a$ and $y = b$. - Transitive: If $(x,y,z) R (a,b,c)$ and $(a,b,c) R (p,q,r)$, then $(x,y,z) R (p,q,r)$. - Since $z = c$ and $c = r$, $z = r$. - From inequalities, $(x - a) \leq (b - y)$ and $(a - p) \leq (q - c)$, adding gives $(x - p) \leq (q - y)$. Thus, $R$ is a partial order. 7. **Totality:** - $R$ is total if for any two elements, either $(x,y,z) R (a,b,c)$ or $(a,b,c) R (x,y,z)$. - Since $z = c$ must hold, if $z \neq c$, neither relation holds. - Therefore, $R$ is not total on $\mathbb{R}^3$. --- 8. **Problem Statement:** In triangle $ACB$ inscribed in a semicircle of radius 2, with diameter $AB = 4$, angle $ACB$ is right. 9. **Using Pythagoras theorem:** $$ a^2 = b^2 + c^2 $$ where $a = AB = 4$. 10. **Explanation:** - Since $AB$ is diameter, angle $ACB$ is right angle by Thales' theorem. - The sides satisfy $4^2 = b^2 + c^2$. **Final answers:** - Exercise 1: Statements true/false as above. - Exercise 2: $R$ is a partial order but not total. - Exercise 3: Right triangle with $a^2 = b^2 + c^2$ and $a=4$.