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 .