\begin{definition}[Definition:Product Notation (Algebra)/Vacuous Product]
Take the composite expressed in product notation:
:$\ds \prod_{\map R j} a_j$
where $\map R j$ is a propositional function of $j$.
Suppose that there are no values of $j$ for which $\map R j$ is true.
Then $\ds \prod_{\map R j} a_j$ is defined as being $1$. '''Beware:''' ''not'' zero.
This composite is called a '''vacuous product'''.
This is because:
:$\forall a: a \times 1 = a$
where $a$ is a number.
Hence for all $j$ for which $\map R j$ is false, the product is unaffected.
This is most frequently seen in the form:
:$\ds \prod_{j \mathop = m}^n a_j = 1$
where $m > n$.
In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$.
\end{definition}