# 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: $$\prod\limits_{i \in S} i^2 = s_1^2 \times s_2^2 \times s_3^2$$.