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I’ve got some questions regarding set theory. I am struggling to find the right notation in order to express a number of conditions. I have a set named A that contains $N$ T-sized groups and each group is characterised by T$B_{x,y}$ elements. For example, consider the following case in which N=4 and T=3.

$$ A:=\{(B_{1,1}, B_{2,1}, B_{3,1}), (B_{3,1}, B_{2,1}, B_{5,1}), (B_{1,3}, B_{2,5}, B_{3,3}), (B_{7,3}, B_{4,5}, B_{3,3})\} $$

Each $A_k$ group (where $1≤k≤N$) represents a specific condition and $P$ is a property which can be found for each condition described by each $A_k$ group (for example, for ($B_{1,1}$,$B_{2,1}$,$B_{3,1}$), $P=5$).

What I want to express is the following conditions:

  1. Find all the P properties for all the $A_k$ groups. Each $P$ property is associated with one $A_k$ group.What I’ve got is: $∀1≤k≤N,P for A_k$
  2. Consider the $A_k$ group(s) which have the smallest value of $P$ property ($P^{min}$) among all P properties.
  3. Consider the $A_k$ groups which contain a specific $B_{x,y}$ element (e.g. $B_{2,1}$). For the above example, $B_{2,1}$ element should be in groups $A_1$ and $A_2$.
  4. I would also like to somehow express the “$P$ for $A_1$ is $5$
  5. Return the $P$ for each of the $A_k$ groups which include the $B_{2,1}$ AND $B_{5,1}$ elements (for the aforementioned example, the $P$ of just $A_2$ group should be returned).

Thanks in advance.

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    I don't understand your indexing scheme for the $B_{x,y}$s. It looks completely random -- and there are two $B_{3,3}$s. Shouldn't it be something like $$A=\{(B_{1,1},B_{1,2},B_{1,3}),(B_{2,1},B_{2,2},B_{2,3}),(B_{3,1},B_{3,2},B_{3,3}),(B_{4,1},B_{4,2},B_{4,3})\}$$ And where do you get $P=5$ from? There doesn't seem to be any relevant $5$s in evidence already.2012-10-27
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    Thanks for your reply. You're right, my $B_{x,y}$ indexing is misleading, yours makes much more sense. Each combination of $B_{x,y}$s which is declared by a corresponding $A_k$ group is statically mapped to a P value. For instance, the combination of $B_{x,y}$s denoted by $A_1$ is mapped to P=5, $A_2$ could be mapped to P=10, etc. I hope it makes more sense now.2012-10-27
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    One minor nitpick: the word "group" has a very technical meaning in mathematics. I think you mean "set" or perhaps "class" every time you use the word "group" in your question.2012-10-28
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    Also, can you give a more concrete example?2012-10-28

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