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Calculus Illustrated -- Notation

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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 $Y$;
  • $i_X:A \hookrightarrow X \quad$ 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: $\vec{v}$.