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Indeed, the "=" sign should probably be substituted with the "element of set" sign there. Both "one dollar" and "one nickel" belong to the set of "things that are money", but two things that belong to the same set aren't required to be the same element of that set. 1 and 2 both belong to the set of natural numbers, but 1 and 2 are not the same natural number.
– probably_someone
Commented Sep 24, 2019 at 11:48
3

@probably_someone Yes. I was going to say that Set-Theory heavily applies here. In all sets with elements, there is some relationship between those elements, and some operators that can work on those sets. Symbols and terms for such operators may look the same, but have different meaning when the set definition is known. Order of operations is relevant in some sets for particular operators as well.
– wolfsshield
Commented Sep 24, 2019 at 14:07
@probably_someone - this may be onto something. It sounds like the solution is to logically disprove the equality of "sameness", or ∃! (*N*∈{Answers}): P(*N'*∈{Questions}) - So in this case, ["Birds" ∈ ({What animals have feathers and can fly?} ⋂ {What was John Audubon's main art subject?})], yet it is false that ⋂⇒=. All {Answers} can belong to multiple {Questions}, they have no uniqueness. This is just the problem restated - not a solution.
– Vogon Poet
Commented Sep 25, 2019 at 20:49
2

@VogonPoet The general class of object that has the important properties of "=" is called an equivalence relation. For a relation ~, we say that ~ is an equivalence relation if it's true that 1) A~A for any A ("~ is reflexive"), 2) A~B implies B~A and vice versa ("~ is symmetric"), and 3) A~B and B~C implies A~C ("~ is transitive"). The subset relation is not symmetric (for example, the even numbers are a subset of the integers, but the integers are not a subset of the even numbers), so it fails criterion 2 of the definition.
– probably_someone
Commented Sep 25, 2019 at 22:19
@VogonPoet That said, the version of "sameness" you're going to get out of an equivalence relation can be quite abstract and broad (which is useful for mathematicians, as it gives them a method of linking seemingly unrelated mathematical objects together, but isn't always useful if you're looking for a way to distinguish between two things that are similar). Choosing the right definition of "the same" is key, and in your case, you want to choose the most restrictive possible definition that still adheres to the three criteria (since you're looking for a tool to make fine-grained distinctions).
– probably_someone
Commented Sep 25, 2019 at 22:25

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