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# Calculus Illustrated -- Notation

This is a partial list of the notation used in Calculus Illustrated.

• $\Longrightarrow \quad$ “therefore”, “then”, “hence”, “only if”, etc.;
• $\Longleftarrow \quad$ “provided”, “if”, etc.;
• $\Longleftrightarrow \quad$ “if and only if” or “is equivalent to”;
• $x\in X\quad$ “$x$ belongs to set $X$” or “$x$ is an element of $X$”;
• $x\not\in X\quad$ “$x$ does not belongs to set $X$” or “$x$ is not an element of $X$”;
• $\forall x \quad$ “for any $x$” or “for each $x$” or “for all $x$”;
• $\exists x\quad$ “there exists (such an) $x$” or “for some $x$”;
• $A\subset B\quad$ set $A$ is a subset of $B$;
• $A \cup B= \{x:\ x\in A\ \texttt{ OR }\ x\in B\}\quad$ the union of sets $A$ and $B$;
• $A \cap B= \{x:\ x\in A\ \texttt{ AND }\ x\in B\}\quad$ the intersection of sets $A$ and $B$;
• $X \setminus B= \{x\in A:\ x\not\in B\}$;
• $f:X\to Y \quad$ a function from set $X$ to set $Y$;
• $i_A:A \hookrightarrow X \quad$ the inclusion function of subset $A\subset X$ of set $X$ into $X$;
• ${\rm Id}_X:X\to X \quad$ the identity function on set $X$;
• $f:x\mapsto y \quad$ function $f:X\to Y$ takes $x\in X$ to $y\in Y$, i.e., $f(x)=y$;
• $\operatorname{Im} f =f(X) \quad$ the (total) image of $f:X\to Y$;
• $\operatorname{Int}(A) \quad$ the interior of subset $A\subset X$ in $X$;
• $\operatorname{Fr}(A) \quad$ the frontier of subset $A\subset X$ in $X$;
• $B(a,\delta)= \{u\in {\bf R}^n :\ ||u-a|| < \delta \} \quad$ the open ball centered at $a$ in ${\bf R}^n$ of radius $\delta$;
• $\bar{B}(a,\delta)= \{u\in {\bf R}^n :\ ||u-a|| \le \delta \} \quad$ the closed ball centered at $a$ in ${\bf R}^n$ of radius $\delta$;
• ${\bf R} \quad$ the real numbers;
• ${\bf C} \quad$ the complex numbers;
• ${\bf Q} \quad$ the rational numbers;
• ${\bf Z} \quad$ the integers;
• ${\bf Z}_n=\{0,1,...,n-1\} \quad$ the integers modulo $n$;
• ${\bf R}^n=\{(x_1,...,x_n):x_i \in {\bf R}\} \quad$ the $n$-dimensional Euclidean space;
• ${\bf R}^n=\{<x_1,...,x_n>:x_i \in {\bf R}\} \quad$ the $n$-dimensional vector space;
• ${\bf B}^n = \{u\in {\bf R}^n: ||u|| \le 1\} \quad$ the closed unit ball in ${\bf R}^n$;
• ${\bf I}=[0,1] \quad$ the closed unit interval;
• ${\bf I}^n \quad$ the $n$-cube in ${\bf R}^n$;
• $x\cdot y \quad$ the dot product.

Notation not used: $\vec{v}$.