Algebra

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