Sum of Summations over Overlapping Domains/Example

Example of Sum of Summations over Overlapping Domains

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

Let $\map R j$ be the propositional function $1 \mathop \le j \mathop \le m$.

Let $\map S j$ be the propositional function $m \mathop \le j \mathop \le n$.

Then we have:

$\ds \map R j \lor \map S j$

$=$

$\ds \paren {1 \mathop \le j \mathop \le m} \lor \paren {m \mathop \le j \mathop \le n}$

$\ds $

$=$

$\ds \paren {1 \mathop \le j \mathop \le n}$

and:

$\ds \map R j \land \map S j$

$=$

$\ds \paren {1 \mathop \le j \mathop \le m} \land \paren {m \mathop \le j \mathop \le n}$

$\ds $

$=$

$\ds \paren {j = m}$

$\blacksquare$