# Algebra

## Quantifiers

### Summations

Given a finite set \(S\) and a function \(f : S \mapsto \mathbb{R}\), \(\sum\limits_{x \in S} f(x)\) represents the sum of all the elements of \(S\) applied to \(f\). Basically, if \(S = \{ s_1, s_2, s_3 \}\), then \(\sum\limits_{x \in S} f(x) = f(s_1) + f(s_2) + f(s_3)\).

Summing up over intervals of integers is so common that there is an special notation for it. Something like \(\sum\limits_{i \in \{ 1, …, 100 \}} i^2\) can be re-expressed as \(\sum\limits_{i = 1}^{100} i^2\). The general form is \(\sum\limits_{i = a}^{b} f(i)\), assuming that \(a \leq b\).

### Products

The product quantifier has the same syntax as the summation quantifier, but of course we calculate the product of every function application. The product of all numbers in a set \(S = \{ s_1, s_2, s_3 \}\) by the power of two is: \(\prod\limits_{i \in S} i^2 = s_1^2 \times s_2^2 \times s_3^2 \).