Sigma notation: what should be the equivalent associative law for product notation?

Question

Below is the associative law for sigma notation.

$$\sum_{k \in K} a_k + \sum_{k \in K} b_k = \sum_{k \in K} (a_k + b_k)$$

What should be the equivalent law for product notation ? and why ?

Should it be

$$\prod_{k \in K} a_k \prod_{k \in K} b_k = \prod_{k \in K} a_k b_k$$ or it should be evaluated for $$\prod_{k \in K} a_k + \prod_{k \in K} b_k$$

Answer 1

I assume we are dealing with a finite product, say $K=\{1,2,\ldots,n\}$. Then $$ \prod_{k\in K}a_k\prod_{k\in K}b_k = (a_1a_2\cdots a_n)(b_1b_2\cdots b_n). $$ Assuming multiplication is commutative, we can write this as $$ (a_1b_1)(a_2b_2)\cdots(a_nb_n) = \prod_{k\in K}a_kb_k. $$

Answer 2

Attention: Your first expression is not a (generalised) associative law for sigma notation.

In the following we consider only finite sums and products to keep things simple and we talk about real numbers only where addition and multiplication is defined and well known.

Let $a_1,a_2,a_3,\ldots,a_m,a_{m+1},a_{m+2},\ldots,a_{n} \in\mathbb{R}$ then the following is valid

\begin{align*} (a_1+a_2)+a_3&=a_1+(a_2+a_3) &\qquad \left(\sum_{k=1}^m a_k\right) + \left(\sum_{k=m+1}^n a_k\right)=\sum_{k=1}^n a_k\tag{1}\\ (a_1\cdot a_2)\cdot a_3&=a_1\cdot(a_2\cdot a_3) &\qquad \left(\prod_{k=1}^m a_k\right)\cdot\left(\prod_{k={m+1}}^n a_k\right)=\prod_{k=1}^n a_k\tag{2} \end{align*}

Line (1) shows the associative law and a generalised associative law of addition. The line (2) shows the corresponding laws when using multiplication.

The idea behind the associative law is that we can group two neighboring elements and add them up (resp. multiply them as in (2)), continue successively this way until all elements are finally condensed in a single element and the result will always be the same irrespectively of the order we do the grouping. But, the order of terms does not change.

Changing order of terms and the consequences is subject to the commutative law.

Conclusion: Given a finite set $K$ and real numbers $a_k\ (k\in K)$ the expressions \begin{align*} \sum_{k \in K} a_k + \sum_{k \in K} b_k = \sum_{k \in K} (a_k + b_k)\qquad &\qquad \prod_{k \in K} a_k \prod_{k \in K} b_k = \prod_{k \in K} a_k b_k \end{align*} are correct, but do not represent an associative law. They are the result of successively applying both, the associative law and the commutative law.