How I can typeset this equation with LaTeX?

Question

I would like to typeset the enclosed equation but placing the last a_{i, j} in the same column as the above a_{i, j}+1 and a_{i, j}-1, and the b_i in the same column as the above b_i's and not as it is presented in this equation, where the last line is just center aligned.

Answer 1

I prefer the cases environment of amsmath. Furthermore, it is advisable to add the word "if" to the conditions:

\documentclass{scrartcl}
\usepackage{amsmath}
\begin{document}

\begin{equation}
  a_{i,j} =
  \begin{cases}
    a_{i,j}+1 & \text{if $b_{i}=1$ and $c_{j}=0$}\\
    a_{i,j}-1 & \text{if $b_{i}=0$ and $c_{j}=1$}\\
    a_{i,j}   & \text{if $b_{i}=0$}
  \end{cases}
\end{equation}
\end{document}

Answer 2

Although I am not exactly sure what you are trying to do (it seems like a circular definition to me), perhaps this answers your question:

\documentclass{article}

\begin{document}
\begin{equation}
a_{i,j}=\left\lbrace
\begin{array}{lll}
a_{i,j}+1, & b_i = 1, & c_j = 0\\
a_{i,j}-1, & b_i = 0, & c_j = 1\\
a_{i,j}  , & b_i = 0  & 
\end{array}
\right.
\end{equation}
\end{document}

Note that you could also use the cases environment from the amsmath package, but in that case you would only have two columns while you seem to need three here (the spacing will be different). A solution with cases would be

\documentclass{article}

\usepackage{amsmath}
\begin{document}
\begin{equation}
  a_{i,j} =
  \begin{cases}
    a_{i,j}+1 & b_i = 1,\quad c_j = 0\\
    a_{i,j}-1 & b_i = 0,\quad c_j = 1\\
    a_{i,j}   & b_i = 0
  \end{cases}
\end{equation}
\end{document}

With output: