For uniformity, I use untraditional notation as exemplified in this table:
| notation | expanded |
|---|---|
| Σi∈{1,2,3}·4×i | 4×1 + 4×2 + 4×3 |
| Πi∈{1,2,3}·4×i | 4×1 × 4×2 × 4×3 |
| ∀i∈{1,2,3}·e>i | e>1 ∧ e>2 ∧ e>3 |
| ∃i∈{1,2,3}·e>i | e>1 ∨ e>2 ∨ e>3 |
We know and accept that an empty sum like Σi∈{}·4×i is 0. What about empty products like Πi∈{}·4×i?
A similar question concerns ∀ and ∃ statements with empty domains.
We use this guiding principle: if we split a sum into two sums, or a product into two products, by splitting the index set into two subsets, we should get the same answer. Examples:
The splitting may also be so one-sided that one subset receives all the original indexes and the other receives none:
In order for this to work in general, we need the empty sum and the empty product to satisfy:
This forces Σi∈{}·4×i to be 0 and Πi∈{}·4×i to be 1.
Similarly, by considering splitting of domains, ∀ and ∃ statements with empty domains should satisfy:
This forces ∀i∈{}·e>i to be true and ∃i∈{}·e>i to be false.
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