\begin{definition}[Definition:Kronecker Delta]
Let $\Gamma$ be a set.
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is the mapping on the cartesian square of $\Gamma$ defined as:
:$\forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} :=
\begin{cases}
1_R & : \alpha = \beta \\
0_R & : \alpha \ne \beta
\end{cases}$
This use of $\delta$ is known as the '''Kronecker delta notation''' or '''Kronecker delta convention'''.
\end{definition}