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I encountered a weird question the other day. Evaluate $\sum_{i\in\emptyset}1$? Is this zero or is it something else?

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    It looks like $0$ to me. – brick Jan 09 '15 at 19:18
  • One way of thinking about this is as $0 \times 1$. The related question of the empty product is not so easy http://math.stackexchange.com/questions/110546/what-is-the-product-of-the-empty-set or http://math.stackexchange.com/questions/1017441/why-is-empty-product-defined-to-be-1. When you have divided the product by every factor, only $1$ is left. – Mark Bennet Jan 09 '15 at 19:44

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It is generally considered as a convention that sums over the empty set are equal to zero and products over the empty set are equal to one.

So you obtain, e.g., $$ \sum_{i \in I \cup \emptyset} a_i = \sum_{i\in I} a_i + \sum_{i \in \emptyset} a_i = \sum_{i \in I} a_i$$ for any collection of real numbers $a_i$, $i \in I$.

Alexandre548
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