1. **Problem Statement:** Given two disjoint sets $A$ and $B$ in a set $S$, find $A \cup B$, $A \cap B$, $A^c$, and $B^c$. 2. **Formulas and Rules:** - Union: $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$ - Intersection: $A \cap B = \{x \mid x \in A \text{ and } x \in B\}$ - Complement: $A^c = S \setminus A$, $B^c = S \setminus B$ - Since $A$ and $B$ are disjoint, $A \cap B = \emptyset$. 3. **Step-by-step:** - Since $A$ and $B$ are disjoint, $A \cap B = \emptyset$. - The union $A \cup B$ contains all elements in $A$ or $B$. - The complement $A^c$ is all elements in $S$ not in $A$. - The complement $B^c$ is all elements in $S$ not in $B$. 4. **Function $f: A \to B$ and statements:** - (a) If $f$ is an isomorphism (تشابهي), then it is bijective and thus a transformation (دالة تحويلية). So statement (a) is **true**. - (b) If $f$ is an isomorphism, then its inverse $f^{-1}$ is also an isomorphism. So statement (b) is **true**. - (c) If $f$ is a transformation (bijective), then it can map onto $B$, so statement (c) is **true**. - (d) There is at least one transformation function from $A$ to $B$ if $|A|=|B|$, so statement (d) is **true**. 5. **Limits:** - $\lim_{n \to +\infty} 7n^2 + 3n + 2 = +\infty$ (dominant term $7n^2$ grows without bound). - $\lim_{n \to +\infty} (-3n^2 + 2) = -\infty$ (dominant term $-3n^2$ goes to negative infinity). - $\lim_{n \to +\infty} \frac{1}{2}n + 3 = +\infty$ (linear growth). - $\lim_{n \to +\infty} 5 - \frac{1}{2}n = -\infty$ (linear term dominates negatively). - $\lim_{n \to +\infty} 3(\frac{1}{2})^n + 7 = 7$ (since $(\frac{1}{2})^n \to 0$). - $\lim_{n \to +\infty} 4(\frac{1}{2})^n + 3 = 3$ (same reasoning). - $\lim_{n \to +\infty} \frac{1}{x}(1 - x^3)$ with $x \to +\infty$: Rewrite as $\lim_{x \to +\infty} \frac{1 - x^3}{x} = \lim_{x \to +\infty} \frac{1}{x} - x^2 = -\infty$. 6. **Verification by expansion:** - For example, $7n^2 + 3n + 2$ grows like $7n^2$ which dominates as $n \to +\infty$. **Final answers:** - $A \cup B$ is the union of $A$ and $B$. - $A \cap B = \emptyset$. - $A^c = S \setminus A$. - $B^c = S \setminus B$. - Statements (a), (b), (c), (d) are all **true**. - Limits: - $\lim_{n \to +\infty} 7n^2 + 3n + 2 = +\infty$ - $\lim_{n \to +\infty} (-3n^2 + 2) = -\infty$ - $\lim_{n \to +\infty} \frac{1}{2}n + 3 = +\infty$ - $\lim_{n \to +\infty} 5 - \frac{1}{2}n = -\infty$ - $\lim_{n \to +\infty} 3(\frac{1}{2})^n + 7 = 7$ - $\lim_{n \to +\infty} 4(\frac{1}{2})^n + 3 = 3$ - $\lim_{x \to +\infty} \frac{1 - x^3}{x} = -\infty$