Definition:Product Notation (Algebra)

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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 $S$.


Definition by Index

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

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


Definition by Inequality

The product of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written:

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


Definition by Propositional Function

Let $R \left({j}\right)$ be a propositional function of $j$.

Then we can write:

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


If more than one propositional function is written under the product sign, they must all hold.


Infinite

Let an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$.

Then the precise meaning of $\displaystyle \prod_{R \left({j}\right)} a_j$ is:

$\displaystyle \prod_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0} } a_j}\right) \times \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$

provided that both limits exist.

If either limit does fail to exist, then the infinite product does not exist.


Index Variable

Consider the product, in either of the three forms:

$\displaystyle \prod_{j \mathop = 1}^n a_j \qquad \prod_{1 \mathop \le j \mathop \le n} a_j \qquad \prod_{R \left({j}\right)} a_j$


The variable $j$, an example of a bound variable, is known as the index variable of the product.


Multiplicand

The set of elements $\left\{{a_j \in S: 1 \le j \le n, R \left({j}\right)}\right\}$ is called the multiplicand.


Notation

The sign $\displaystyle \prod$ is called the product sign and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of product.


Vacuous Product

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


Historical Note

The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.