Let \(S=(1,2,+\(ldots+12\)) and \(T_1,T_2+\(ldots+T_a\).
I think you intended $\displaystyle S= (1, 2, +(\ldots+ 12))$ and $\displaystyle (T_1, T_2+ (\ldots+ T_a))$
but I have no idea what thatis supposed to mean! I think those are intended to be sets but what does the "+" mean? And what are you trying to say about them?
As written, that post really has no clear meaning. I suspect you are using some sort of translation program.
Moreover, this must be part of a larger question. You should post the whole question.
It says that $\displaystyle T_a\subseteq T$ but says nothing about how $\displaystyle T_a\text{ nor }T$.
I personally have never the term $\displaystyle Sly$. Please define.
No one criticized you.
The posted question is unreadable in English.
It does not define very essential parts of the question.
The fact that it "clearly mentioned that $\displaystyle T_1,T_2,\cdots, T_a$ are subsets of $\displaystyle S$", tells us nothing whatsoever about their properties.
No one can help with a totally unreadable question.
Let T be a collection of sets $\displaystyle T_i$ for which Ti⊄Tj∀i≠j. If two different subsets have the same cardinality they are only subsets of each other iff they are equal. Define $\displaystyle T_{(k)}$ to be a collection of subsets of size k (for which no two are subsets of each other). Then in this case, $\displaystyle |T_{(6)}| = {12 \choose 6} = 924$ is a maximum for sets of equal size (you can verify that sets of smaller size give a smaller collection). Now, suppose that we "mix" sets of different sizes to attain a larger number. For every set $\displaystyle T_i$ of size n that we include in our set T, we cannot include $\displaystyle {12 \choose (n-1)}$ subsets in our collection (this would violate the condition Ti⊄Tj above). Specifically, if we introduced a larger set in $\displaystyle T_{(6)}$ to make it "bigger" it would necessarily be a superset to a subset in $\displaystyle T_{(6)}$ (hence violating the condition).
@majamin, that answer makes perfect sense. But how did you get that from tbhe other postings?
@geniusgarvil, I understand that you may have a language problem, a translation problem.
But if majamin read your question correctly, why in the world did you not post it clearly?
"What is the maximal number of subsets of S that no two have the property that neither is a subset of the other?"