Definition:Kronecker Delta

Definition
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 defined as:


 * $\forall \left({\alpha, \beta}\right) \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.

It can be expressed in Iverson bracket notation as:
 * $\delta_{\alpha \beta} := \left[{\alpha = \beta}\right]$