Summation over Union of Disjoint Finite Index Sets

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Theorem

Let $\struct{G, +}$ be a commutative monoid.

Let $I$ and $J$ be disjoint finite indexing sets.

Let $K = I \cup J$.

Let $\family{g_k}_{k \mathop \in K}$ be an indexed family of elements of $G$.

Then:

$\ds \sum_{k \mathop \in K} g_k = \paren{\sum_{i \mathop \in I} g_i} + \paren{\sum_{j \mathop \in J} g_j}$

where:

$\ds \sum_{k \mathop \in K} g_k$ is the summation of $g$ over $K$

$\ds \sum_{i \mathop \in I} g_i$ is the summation of the restriction of $g$ over $I$

$\ds \sum_{j \mathop \in J} g_j$ is the summation of the restriction of $g$ over $J$

Proof

Let $\set{i_1, i_2, \ldots, i_n}$ be an enumeration of $I$.

Let $\set{j_1, j_2, \ldots, j_m}$ be an enumeration of $J$.

Let $k: \closedint 1 {n+m}$ be the mapping defined by:

$k_l = \begin{cases}

i_l & : \text{ if } 1 \le l \le n \\ j_{l-n} & : \text{ if } l > n \\ \end{cases}$

From Union of Bijections with Disjoint Domains and Codomains is Bijection

$\set{k_1, k_2, \ldots, k_{n + m}}$ is an enumeration of $K$

We have:

\(\ds \sum_{k \mathop \in K} g_k\)

\(=\)

\(\ds \sum_{l \mathop = 1}^{n \mathop + m} g_{k_l}\)

Definition of Summation over Finite Index

\(\ds \)

\(=\)

\(\ds \paren{\sum_{l \mathop = 1}^n g_{k_l} } + \paren{\sum_{l \mathop = n \mathop + 1}^{n \mathop + m} g_{k_l} }\)

Monoid Axiom $\text S 1$: Associativity

\(\ds \)

\(=\)

\(\ds \paren{\sum_{l \mathop = 1}^n g_{i_l} } + \paren{\sum_{l \mathop = n \mathop + 1}^{n \mathop + m} g_{j_{l \mathop - n} } }\)

definition of $k$

\(\ds \)

\(=\)

\(\ds \paren{\sum_{l \mathop = 1}^n g_{i_l} } + \paren{\sum_{l \mathop = 1 }^m g_{j_l} }\)

re-indexing second summation

\(\ds \)

\(=\)

\(\ds \paren{\sum_{i \mathop \in I} g_i } + \paren{\sum_{j \mathop \in J} g_j}\)

Definition of Summation over Finite Index

$\blacksquare$