Typesetting displayed equivalences and definitions of unequal length

Question

What is the proper way to typeset in LaTeX, in display-mode, a series of equivalences and definitions of unequal lengths of sets ?

Consider for this the following code that I used for a series of equivalences

$$
\begin{array}{rcl}
y_{0}=\frac{\pm\sqrt{1-z^{2}}}{z} & \Longleftrightarrow & y_{0}z=\pm\sqrt{1-z^{2}}\\
 & \Longleftrightarrow & y_{0}^{2}z^{2}=1-z^{2}\\
 & \Longleftrightarrow & (1+y_{0})z^{2}=1\\
 & \Longleftrightarrow & z=\pm\frac{1}{1+y_{0}^{2}}.
\end{array}
$$

and the following code I used for definitions of unequal lengths of sets:

\begin{alignat*}{1}
D_{1} & :=C_{a}\cap\left\{ (x,y,z)\in D_{2}:x>0\right\} \ \text{and}\\
D_{2} & :=\left\{ (x,y,z)\in D_{1}:x<0\right\} .
\end{alignat*}

(Ignore the mathematical non-sense.) Are the environments I used as well as their alignments the proper ones for this ?

Answer 1

I'd use align* environment.

\begin{align*}
y_{0}=\frac{\pm\sqrt{1-z^{2}}}{z} &\Longleftrightarrow  y_{0}z=\pm\sqrt{1-z^{2}}\\
                                  &\Longleftrightarrow  y_{0}^{2}z^{2}=1-z^{2}\\
                                  &\Longleftrightarrow  (1+y_{0})z^{2}=1\\
                                  &\Longleftrightarrow  z=\pm\frac{1}{1+y_{0}^{2}}.
\end{align*}

\begin{align*}
D_{1} & :=C_{a}\cap\left\{ (x,y,z)\in D_{2}:x>0\right\} \text{ and}\\
D_{2} & :=\left\{ (x,y,z)\in D_{1}:x<0\right\} .
\end{align*}

ps. \left\{ \right\} is not necessary in your case. Simply use \{ \} or some variant \bigl\{ \bigr\} or \Bigl\{ \Bigr\}.

Answer 2

You have another possibility with the mathtools package: it defines an \ArrowBetweenLines command which places by default an \Updownarrow between the lines of an alignat environment, on the left. There is a starred version which places it on the right.

It accepts as an optional argument the symbol that will be used, and I took advantage of this feature to define a \TextBetweenLines command, that I used in the second example in the O.P.'s question:

\documentclass{article}
\usepackage{mathtools}
\newcommand\TextBetweenLines[1]{\ArrowBetweenLines[\text{#1}]}

\begin{document}

\begin{alignat*}{2}
                     & & y_{0} & =\frac{\pm\sqrt{1-z^{2}}}{z} \\
  \ArrowBetweenLines & & y_{0}z & =\pm\sqrt{1-z^{2}} \\
  \ArrowBetweenLines & & y_{0}^{2}z^{2} & =1-z^{2}
\end{alignat*}

\begin{alignat*}{2}
  y_{0}^{2}z^{2} & =1-z^{2} & & \\
  \ArrowBetweenLines *(1+y_{0})z^{2} & =1 & & \\
  \ArrowBetweenLines* z & =\pm\frac{1}{1+y_{0}^{2}}. & &
\end{alignat*}

\begin{alignat*}{2}
                         & & D_{1} & \coloneqq C_{a}\cap\left\{ (x,y,z)\in D_{2}:x>0\right\} \\
  \TextBetweenLines{and} & & D_{2} & :=\left\{ (x,y,z)\in D_{1}:x<0\right\} .
\end{alignat*}

\end{document}