Math and science::Analysis::Tao::07. Series

# Series laws Ⅱ: sums over finite sets

9 basic properties of sums over finite sets. The proofs of these are a good exercise to refresh the notions of functions, bijections, sets, sums and induction.

There is 1-1 correspondence between many of these properties and the 6 properties already shown for sums of the form $$\sum_{i=1}^{n}a_i$$. Thus, proving many of them amounts to choosing a bijection that allows transitioning to one of the original properties.

Question: when reading the below properties, consider:

• what is the equivalent $$\sum_{i=1}^{n}a_i$$ form, if it exists?
• does the property extend to sums over infinite sets?

1. [...]. If $$X$$ is empty, and $$f: X \rightarrow \mathbb{R}$$ is a function (i.e., $$f$$ is the empty function), we have

$\sum_{x \in X} f(x) = 0$

2. [...]. If $$X$$ consists of a single element, $$X = \{x_0\}$$, and $$f: X \rightarrow \mathbb{R}$$ is a function, we have

$\sum_{x \in X}f(x) = f(x_0)$

3. [...], part I. If $$X$$ is a finite set, $$f: X \rightarrow \mathbb{R}$$ is a function, and $$g: Y \rightarrow X$$ is a bijection, then

$\sum_{x \in X}f(x) = \sum_{y \in Y}f(g(y))$

4. [...], part II. Let $$n \le m$$ be integers, and let $$X$$ be the set $$X := \{i \in \mathbb{Z} : n \le i \le m\}$$. If $$a_i$$ is a real number assigned to each integer $$i \in X$$, then we have

$\sum_{i=n}^{m}a_i = \sum_{i \in X}a_i$

5. [...]. Let $$X$$, $$Y$$ be disjoint finite sets ($$X \cap Y = \emptyset$$), and $$f: X \cup Y \rightarrow \mathbb{R}$$ is a function. Then we have

$\sum_{z \in X \cup Y}f(z) = \left( \sum_{x \in X} f(x) \right) + \left( \sum_{y \in Y} f(y) \right)$

6. [...], part I. Let $$X$$ be a finite set, and let $$f: X \rightarrow \mathbb{R}$$ and $$g: X \rightarrow \mathbb{R}$$ be functions. Then we have

$\sum_{x \in X}(f(x) + g(x)) = \sum_{x \in X}f(x) + \sum_{x \in X}g(x)$

7. [...], part II. Let $$X$$ be a finite set, and let $$f: X \rightarrow \mathbb{R}$$ be a function, and let $$c$$ be a real number. Then

$\sum_{x \in X}cf(x) = c\sum_{x \in X}f(x)$

8. [...]. Let $$X$$ be a finite set, and let $$f: X \rightarrow \mathbb{R}$$ and $$g: X \rightarrow \mathbb{R}$$ be functions such that $$f(x) \le g(x) \text{ for all } x \in X$$. Then

$\sum_{x \in X}f(x) \le \sum_{x \in X}g(x)$

9. [...]. Let $$X$$ be a finite set, and let $$f: X \rightarrow \mathbb{R}$$ be a function, then

$|\sum_{x \in X}f(x)| \le \sum_{x \in X}|f(x)|$