I have a problem with subsets in Set Theory.
a set A is a subset of set B if all the elements of A are also elements of B.
{1,2} is a subset of {1,2,3}
Simple enough.
But, as I understand it, the word "subset" means a set within another set.
So, intuitively if A is in B It would mean that B is {{1,2},3}
This would mean that {1,2,3} and {{1,2},3} and {1,{2,3}} are the same set as brackets would be arbitrary ways of grouping elements in subsets within the set.
This is generally the way we understand and draw sets as circles within circles. Circles are drawn depending on how we want to group the content but they have no substance in and on themselves.
This is also backed up by the fact that two identical elements are the same element, and by that token, subset A is entirely inside the set B.
But when looking at cardinalities, it seems those 3 sets are very different as {1,2,3} has a cardinality of 3 while {{1,2},3} and {1,{2,3}} have a cardinality of 2.
If they were the same set, their cardinalities would be ambiguous.
Therefore, they aren't the same set.
So, I conclude that I am wrong in the way I see subsets as subsets are NOT sets within sets, they are just sets which content is also included in another set.
Right?
In a nutshell, it's hard to tell if a set is only its elements or if a set is the collection of the elements AND the brackets around it.
It can't be more basic than that but I'm confused since both interpretations could make sense but they are incompatible.
If brackets are included a subset is not a set within a set (since it would make cardinality inconsistent), and if brackets are not included how can cardinality depend on sets within sets?
It's not just a problem of semantics since I get actual inconsistencies depending on how I understand it
No matter how I see it, I can't seem to reconcile subsets and cardinality.
EDIT: a user completely edited the title but the new formulated question was not the question I asked (and some things I considered important in my text had been erased too). So I decided to rollback to what it was.
Thank you very much though for editing the format. If needed I'm happy to edit anything myself, if I'm asked to. My username can only be associated with my own words. It seems fair.
Best Answer
Brackets do make a huge difference in Math!
There is a fundamental difference between $A\subset B$ ($A$ is a subset of $B$) and $A\in B$ ($A$ is a member of $B$).
The former means that any member of $A$ is a member of $B$, while the latter means that $A$ itself is a member of $B$. This said $$ \{1,2\} \subset \{1,2,3\} $$ but $$ \{1,2\} \not\in\{1,2,3\} $$ while $$ \{1,2\} \not\subset \{\{1,2\},3\} $$ but $$ \{1,2\} \in\{\{1,2\},3\}. $$