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:
 * $$\delta_{\alpha \beta} = \left[{\alpha = \beta}\right]$$