Thread: Sly Subsets of S

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?

Actually, the question goes like this.
Let S={1,2,3,…12} and T1,T2,…Ta be subsets of S such that Ti⊄Tj∀i≠j. What is the maximum possible value of a?

Originally Posted by geniusgarvil

Actually, the question goes like this.
Let S={1,2,3,…12} and T1,T2,…Ta be subsets of S such that Ti⊄Tj∀i≠j. What is the maximum possible value of a?

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.

As i have mentioned the complete question,and also clearly mentioned that T1,T2,.... Ta are subsets of S, if u are not able to solve it , u are not allowed to criticize me.

Originally Posted by geniusgarvil

As i have mentioned the complete question,and also clearly mentioned that T1,T2,.... Ta are subsets of S, if u are not able to solve it , u are not allowed to criticize me.

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.

Originally Posted by geniusgarvil

Actually, the question goes like this.
Let S={1,2,3,…12} and T1,T2,…Ta be subsets of S such that Ti⊄Tj∀i≠j. What is the maximum possible value of a?

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).

Originally Posted by majamin

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).

@majamin, that answer makes perfect sense. But how did you get that from tbhe other postings?

Originally Posted by geniusgarvil

Let \(S=(1,2,+\(ldots+12\)) and \(T_1,T_2+\(ldots+T_a\).

@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?"

Originally Posted by geniusgarvil

Actually, the question goes like this.
Let S={1,2,3,…12} and T1,T2,…Ta be subsets of S such that Ti⊄Tj∀i≠j. What is the maximum possible value of a?

Originally Posted by Plato

@majamin, that answer makes perfect sense. But how did you get that from tbhe other postings?

The initial post was nonsensical, but when geniusgarvil reposted it (above quote) it made enough sense to answer, I thought.