Definition:Product Notation (Algebra)/Vacuous Product

Definition
Take the composite expressed in product notation:
 * $\displaystyle \prod_{R \left({j}\right)} a_j$

where $R \left({j}\right)$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $R \left({j}\right)$ is true.

Then $\displaystyle \prod_{R \left({j}\right)} a_j$ 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 $\Phi \left({j}\right)$ 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

 * Definition:Vacuous Truth
 * Definition:Vacuous Summation