Math Notations 358Fbd

1. The user has provided a list of mathematical notations and definitions related to sets, intervals, functions, graphs, norms, open sets, and convex sets. 2. Since no specific problem statement or question is given, there is no explicit problem to solve. 3. The content mainly defines: - Sets of numbers: $\mathbb{N}$, $\mathbb{N}^*$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{R}^+$, $\mathbb{C}$ - Intervals of integers: $[a,b] = \{a, a+1, ..., b\}$ for integers $a \leq b$ - Set inclusion: $A \subseteq B$, $A \subset B$, cardinality $|A|$ - Functions: $f : A \to B$, image $f(X) = \{f(x) : x \in X\}$ - Graphs: $G = (V,E)$ with vertices $V$ and edges $E$, cardinalities $|V|$, $|E|$ - Norm of a vector $x = (x_1, ..., x_E) \in \mathbb{R}^E$ defined as $$||x|| = \sqrt{x_1^2 + ... + x_E^2}$$ - Open sets in $\mathbb{R}^E$ defined via open balls $B(x,r) = \{y \in \mathbb{R}^E : ||y - x|| < r\}$ - Convex sets $C \subseteq \mathbb{R}^E$ where for all $x,y \in C$ and $t \in [0,1]$, the convex combination $tx + (1-t)y \in C$ 4. Without a specific question, no calculation or solution can be provided. 5. Please provide a specific problem or question to solve.