Alignment building blocks

Some other equation alignment environments, such as aligned and gathered, construct self-contained units that can be used inside of other expressions, or set side-by-side. These environments take an optional argument to specify their vertical positioning with respect to the material on either side. The default is `middle' placement with the vertical midpoint of the total unit falling on the math axis

. For example:

\begin{equation*}
\begin{aligned}
\alpha&=\alpha\alpha\\ 
\beta&=\beta\beta\beta\beta\beta\\ 
\gamma&=\gamma
\end{aligned}
\qquad\text{versus}\qquad
\begin{aligned}[t]
\delta&=\delta\delta\\ 
\eta&=\eta\eta\eta\eta\eta\eta\\ 
\varphi&=\varphi
\end{aligned}
\end{equation*}

``Cases'' constructions like the following are common in mathematics:

and in the amsmath package there is a cases environment to make them easy to write:

P_{r-j}=\begin{cases}
    0&  \text{if $r-j$ is odd},\\ 
    r!\,(-1)^{(r-j)/2}&  \text{if $r-j$ is even}.
  \end{cases}