Sum of Summations over Overlapping Domains

Theorem

Let $R: \Z \to \set {\T, \F}$ and $S: \Z \to \set {\T, \F}$ be propositional functions on the set of integers.

Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.

Let the fiber of truth of both $R$ and $S$ be finite.

Then:

$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j = \sum_{\map R j \mathop \lor \map S j} a_j + \sum_{\map R j \mathop \land \map S j} a_j$

where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.

Let the fiber of truth of both $R$ and $S$ be infinite.

Then provided that any $3$ of the $4$ summations converge:

$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j = \sum_{\map R j \mathop \lor \map S J} a_j + \sum_{\map R j \mathop \land \map S j} a_j$

where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.

$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j$

$=$

$\ds \sum_{j \mathop \in \Z} a_j \sqbrk {\map R j} + \sum_{j \mathop \in \Z} a_j \sqbrk {\map R j}$

$\ds $

$=$

$\ds \sum_{j \mathop \in \Z} a_j \paren {\sqbrk {\map R j} + \sqbrk {\map R j} }$

Let:

$A := \set {j \in \Z: \map R j}$

$B := \set {j \in \Z: \map R j}$

The result then follows from Cardinality of Set Union.

$\blacksquare$

$\ds \sum_{1 \mathop \le j \mathop \le m} a_j + \sum_{m \mathop \le j \mathop \le n} a_j = \paren {\sum_{1 \mathop \le j \mathop \le n} a_j} + a_m$