Definition:Product Notation (Algebra)

Definition

Let $\left({S, \times}\right)$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Then the composite is called the product of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:

$\displaystyle \prod_{j=1}^n a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

Alternatively:

$\displaystyle \prod_{1 \le j \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:

$\displaystyle \prod_{\Phi \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

Multiplicand

The quantity after the product sign is called the multiplicand, or the set of multiplicands.

Vacuous Product

Take the product:

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

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

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

Then $\displaystyle \prod_{\Phi \left({j}\right)} a_j$ is defined as being $1$. Beware: not zero.

This summation is called a vacuous product.

This is most frequently seen in the form:

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

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 $n$.

Compare vacuous truth.

Cartesian Product of Sets

The following notation is also customary.

Let $\left \langle {S_n} \right \rangle$ be a sequence of sets.

The cartesian product of $\left \langle {S_n} \right \rangle$ can be written as:

$\displaystyle \prod_{k=1}^n S_k = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S_k}\right\}$

Also see

• Sum notation

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 18$
• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.3 \ (20)$