Even I try but I can not understand the following equation:
$ Poss(A) = \bigcap_{a \in A}\lbrace e | e \in Poss(a) \rbrace $
where $A=\lbrace a1,a2,a3 \rbrace$ and $Poss(a)$ is a set of sets, for example:
$Poss(a1)=\lbrace \lbrace 2,7 \rbrace, \lbrace 1,2,5,7 \rbrace \rbrace $
$Poss(a2)=\lbrace \lbrace 3,6 \rbrace, \lbrace 1,3,6 \rbrace \rbrace $
$Poss(a3)=\lbrace \lbrace 4,6 \rbrace, \lbrace 1,4,6 \rbrace \rbrace $
So, I am not sure which one of the following two is right? Or both are wrong?
$Poss(A)= \emptyset$ because there is no common sets between Poss(a1) Poss(a2), and Poss(a3)
OR
$Poss(A)= \lbrace 1 \rbrace$ because 1 in $\lbrace 1,2,5,7 \rbrace $ of Poss(a1), in $\lbrace 1,3,6 \rbrace $ of Poss(a2), and in $\lbrace 1,4,6 \rbrace $ of Poss(a3).
Please help me.
Thank you very much.
Since $\{e\mid e\in\text{Poss}(a)\}=\text{Poss}(a)$ you might as well write: $$\text{Poss}(A)=\bigcap_{a\in A}\text{Poss}(a)$$
So actually we have:$$e\in\text{Poss}(A)\iff\forall a\in A[e\in\text{Poss}(a)]$$
So if $A=\{a_1,a_2,a_3\}$ this comes to the same as:$$e\in\text{Poss}(A)\iff [e\in\text{Poss}(a_1)\wedge e\in\text{Poss}(a_2)\wedge e\in\text{Poss}(a_2)]\tag1$$
Can you find any set $e$ that satisfies the RHS of $(1)$?