1. **Problem Statement:** (a) Determine if each function $f_i : X \times X \to \mathbb{R}$ with $X=\{a,b,c,d\}$ given by the tables $f_1, f_2, f_3, f_4$ is a metric. If not, identify violated axioms. (b) Prove uniform continuity of $f:\mathbb{R}^n \to \mathbb{R}$ defined by $f(x_1,...,x_n) = \sum_{i=1}^n i x_i$ under metric $d(x,y) = \sum_{i=1}^n |x_i - y_i|$. (c) For subsets $A=(0,2) \cup [3,5)$ and $B=(-\infty,-1) \cup (1,\infty)$ in $\mathbb{R}$, find boundary, interior, and closure. 2. **Metric Definition and Axioms:** A function $d:X\times X \to \mathbb{R}$ is a metric if for all $x,y,z \in X$: - (Non-negativity) $d(x,y) \geq 0$ - (Identity) $d(x,y)=0$ iff $x=y$ - (Symmetry) $d(x,y) = d(y,x)$ - (Triangle inequality) $d(x,z) \leq d(x,y) + d(y,z)$ 3. **Check $f_1$:** - Symmetry: Check $f_1(b,c)=1$ vs $f_1(c,b)=5$ (not equal) → violates symmetry. - Identity: $f_1(a,a)=0$, etc. holds. - Triangle inequality: Not checked since symmetry fails. **Conclusion:** $f_1$ is not a metric; violates symmetry. 4. **Check $f_2$:** - Symmetry: $f_2(b,c)=1$, $f_2(c,b)=2$ (not equal) → violates symmetry. **Conclusion:** $f_2$ is not a metric; violates symmetry. 5. **Check $f_3$:** - Symmetry: $f_3(b,c)=1$, $f_3(c,b)=0$ (not equal) → violates symmetry. **Conclusion:** $f_3$ is not a metric; violates symmetry. 6. **Check $f_4$:** - Symmetry: $f_4(a,b)=2$, $f_4(b,a)=2$ (equal), check others similarly. - Identity: $f_4(a,a)=0$, etc. holds. - Triangle inequality: Check sample: $f_4(a,c)=1 \leq f_4(a,b)+f_4(b,c)=2+2=4$ holds. **Conclusion:** $f_4$ satisfies metric axioms; $f_4$ is a metric. 7. **Part (b) Uniform Continuity:** - Given $d(x,y) = \sum_{i=1}^n |x_i - y_i|$ and $f(x) = \sum_{i=1}^n i x_i$. - For $x,y \in \mathbb{R}^n$, $$|f(x)-f(y)| = \left| \sum_{i=1}^n i(x_i - y_i) \right| \leq \sum_{i=1}^n i |x_i - y_i| \leq n \sum_{i=1}^n |x_i - y_i| = n d(x,y).$$ - Choose $\delta = \varepsilon / n$ for any $\varepsilon > 0$. - Then if $d(x,y) < \delta$, $|f(x)-f(y)| < \varepsilon$. - Hence, $f$ is uniformly continuous on $\mathbb{R}^n$. 8. **Part (c) Boundary, Interior, Closure:** - For $A = (0,2) \cup [3,5)$: - Interior $\mathrm{Int}(A) = (0,2) \cup (3,5)$ (open parts inside). - Closure $\overline{A} = [0,2] \cup [3,5]$ (include limit points). - Boundary $\partial A = \overline{A} \setminus \mathrm{Int}(A) = \{0,2,3,5\}$. - For $B = (-\infty,-1) \cup (1,\infty)$: - Interior $\mathrm{Int}(B) = (-\infty,-1) \cup (1,\infty)$ (already open). - Closure $\overline{B} = (-\infty,-1] \cup [1,\infty)$. - Boundary $\partial B = \{-1,1\}$. **Final answers:** - $f_1, f_2, f_3$ are not metrics (violate symmetry). - $f_4$ is a metric. - $f$ is uniformly continuous. - Sets $A,B$ boundaries, interiors, closures as above.