Definition:Product Notation (Algebra)/Vacuous Product

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Definition

Take the composite expressed in product notation:

$\displaystyle \prod_{\map R j} a_j$

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 $\displaystyle \prod_{\map R j} a_j$ 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:

$\displaystyle \prod_{j \mathop = m}^n a_j = 1$

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 $n$.


Also see


Linguistic Note

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.


Sources