Basic notes on algebra



Given a finite set SS and a function f:Sf : S \mapsto \mathbb{R}, xSf(x)\sum\limits_{x \in S} f(x) represents the sum of all the elements of SS applied to ff. Basically, if S={s1,s2,s3}S = \{ s_1, s_2, s_3 \}, then xSf(x)=f(s1)+f(s2)+f(s3)\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 i{1,...,100}i2\sum\limits_{i \in \{ 1, ..., 100 \}} i^2 can be re-expressed as i=1100i2\sum\limits_{i = 1}^{100} i^2. The general form is i=abf(i)\sum\limits_{i = a}^{b} f(i), assuming that aba \leq b.


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={s1,s2,s3}S = \{ s_1, s_2, s_3 \} by the power of two is: iSi2=s12×s22×s32\prod\limits_{i \in S} i^2 = s_1^2 \times s_2^2 \times s_3^2.