Sums and Products over a Finite Set and over an Interval

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\ast$ be an associative and commutative binary operation on a set $X$, let $A$ be a finite set, and let $f:A\to X$. Let $\#$ be as in The Number of Elements of a Finite Set §cardinality, intervals and $[n]$ as in Intervals of Natural Numbers §interval and Intervals of Natural Numbers §segment, and iterated operations along $[n]$ as in Iterated Operations: Finite Sums and Finite Products §iterated.

¶ If $A\neq\emptyset$, then

$$\mathop{\ast}\limits_{x\in A}f(x)=\mathop{\ast}\limits_{k=1}^{\#A}f(\varphi(k))$$

for any bijection $\varphi$ from $[\#A]$ onto $A$; here $\#A\in\mathbb{N}$ by The Iterated Operation over a Finite Set Does Not Depend on the Enumeration §nonempty, such a bijection exists by The Number of Elements of a Finite Set §cardinality, and the value does not depend on it by The Iterated Operation over a Finite Set Does Not Depend on the Enumeration §independent.

¶ If $A=\emptyset$ and $\ast$ has a neutral element $e$, which is unique by A Binary Operation Has at Most One Neutral Element §unique, then $\mathop{\ast}\limits_{x\in A}f(x)=e$.

¶ Let $A\neq\emptyset$ or let $\ast$ have a neutral element. For a map $f$ from a set containing $A$ to $X$, $\mathop{\ast}\limits_{x\in A}f(x)$ denotes the iterated operation of $f|_{A}$. For a property $P$, $\mathop{\ast}\limits_{x\in A,\,P(x)}f(x)$ denotes that over $\{x\in A:P(x)\}$, which is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §subset, provided this set is nonempty or $\ast$ has a neutral element. For finite sets $A$ and $B$ and $f:A\times B\to X$, the iterated operation over $A\times B$, which is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §union, is also written $\mathop{\ast}\limits_{(x,y)\in A\times B}f(x,y)$.

¶ For $m,n\in\mathbb{N}_{0}$ and a map $a$ from a set containing $\{m,\dots,n\}$ to $X$, $\mathop{\ast}\limits_{k=m}^{n}a_{k}=\mathop{\ast}\limits_{k\in\{m,\dots,n\}}a_{k}$, the interval being finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §naturals, provided $m\le n$ or $\ast$ has a neutral element. For $m=1\le n$ this agrees with Iterated Operations: Finite Sums and Finite Products §iterated, by taking $\varphi=\mathrm{id}_{[n]}$, as $\#[n]=n$ by Counting: Intervals, Empty Sets and Singletons, Injective Images, Subsets, Unions and Products §intervals.

¶ If $\ast$ is written $+$, these are written $\sum_{x\in A}f(x)$, $\sum_{x\in A,\,P(x)}f(x)$ and $\sum_{k=m}^{n}a_{k}$ and called sums; if $\ast$ is written $\cdot$, they are written with $\prod$ and called products.