Comment by vpetrovykh
7 years ago
The sum of an empty set is, in fact, 0 (the identity for addition).
The generalized conjuction (we have a function called "all" for that) of an empty set is True (the identity for conjunction).
The generalized disjuction (we have a function called "any" for that) of an empty set is False (the identity for disjunction).
All of the above "sum", "all", and "any" are basically aggregate functions that operate on sets as a whole.
There is no special logic that you wouldn't get from considering these operations generalized for a set.
Thanks, that was helpful. However, I still think that having "sum(1,1,{})" returning "2" and "1+1+{}" returning "{}" can be viewed as somewhat inconsistent.
Let me make a tiny correction to the expression you wrote:
"sum({1, 1, {}})" - the function sum takes only one argument and it's a set. Because we flatten all "nested" sets, the expression "{1, 1, {}}" is equivalent to "{1} UNION {1} UNION {}".
The expression "1 + 1 + {}" albeit valid grammatically, can be equivalently re-written as "{1} + {1} +{}". At this point it should be far more obvious why "sum({1} UNION {1} UNION {})" is not the same as "{1} + {1} + {}".
Literals may be a little confusing because they look like elements, but they are still sets, singleton sets, specifically. There's practical value in simply thinking about "a bunch of things: A, B, C", where each of the A, B and C can themselves be empty, a single thing, or a bunch of things while ignoring nesting. In our case we allow duplication in these bunches (which is not part of the bunch theory: http://www.cs.toronto.edu/~hehner/bunch.pdf). However, because most people are familiar with sets we find it easier to keep using the terms "set" and "multi-set" (and stipulate that they are flattened) in explanations.
In general, the way the operator "+" works is this: A + B = {a + b : for all a in A, for all b in B}. Whereas the expression "{A, B}" is defined to be equivalent to "A UNION B".
Good explanation! (which indicates why the handling of empty sets is sometimes a bit confusing)
One more question: What was the motivation behind defining "sum({})" to be "0" rather then "{}" ?
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