Indexed Summation of Sum of Mappings
Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f, g: \closedint a b \to \mathbb A$ be mappings.
Let $h = f + g$ be their pointwise sum.
Then we have the equality of indexed summations:
- $\ds \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$
Proof
The proof proceeds by induction on $b$.
For all $b \in \Z_{\ge 0}$, let $\map P b$ be the proposition:
- $\ds \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$
Basis for the Induction
Let $b < a$.
Then all indexed summations are zero.
Because $0 = 0 + 0$, the result follows.
Basis for the Induction
Let $b = a$.
| \(\ds \sum_{i \mathop = a}^b \map h i\) | \(=\) | \(\ds \sum_{i \mathop = a}^a \map h i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map h a\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map f a + \map g a\) | Definition of Pointwise Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^a \map f i + \sum_{i \mathop = a}^a \map g i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i\) | as $a = b$ |
Thus $\map P a$ is seen to hold.
This is our basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge a$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\ds \sum_{i \mathop = a}^k \map h i = \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i$
from which it is to be shown that:
- $\ds \sum_{i \mathop = a}^{k + 1} \map h i = \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i$
Induction Step
We have:
| \(\ds \sum_{i \mathop = a}^{k + 1} \map h i\) | \(=\) | \(\ds \sum_{i \mathop = a}^k \map h i + \map h {k + 1}\) | Definition of Indexed Summation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^k \map h i + \map f {k + 1} + \map g {k + 1}\) | Definition of Pointwise Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i + \map f {k + 1} + \map g {k + 1}\) | Induction Hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^k \map f i + \map f {k + 1} + \sum_{i \mathop = a}^k \map g i + \map g {k + 1}\) | Commutative Law of Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i\) | Definition of Indexed Summation |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
$\blacksquare$