TAOCP 1.2.3 Exercise 23
Defining $\sum_{R(j)} a_j = 0$ when no integers satisfy $R(j)$ preserves the additive identity.
Section 1.2.3: Sums and Products
Exercise 23. [10] Explain why it is a good idea to define $\sum_{R(j)} a_j$ and $\prod_{R(j)} a_j$ as zero and one, respectively, when no integers satisfy $R(j)$.
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Defining $\sum_{R(j)} a_j = 0$ when no integers satisfy $R(j)$ preserves the additive identity. That is, if we consider an arbitrary sum $\sum_{R(j)} a_j + \sum_{S(j)} a_j$ and some value of $j$ belongs to neither $R$ nor $S$, adding zero does not alter the total. This convention allows the algebraic laws of sums, such as distributivity and rule (d) for manipulating domains, to hold uniformly without requiring special cases for empty index sets.
Similarly, defining $\prod_{R(j)} a_j = 1$ when no integers satisfy $R(j)$ preserves the multiplicative identity. If a product is over an empty set, taking the value 1 ensures that multiplying by this product does not change the value of other factors, and that formulas analogous to sums, including distributive laws for products, remain valid without exceptions. These definitions guarantee uniformity and simplicity in manipulating sums and products in general expressions.
This completes the proof.
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