I would like to ask about notation, because I think that I am missing something.
It is from book; "The Joy of Sets" - Keith Devlin. There is presented an alternative notation; $ x = \{a | P(a) \} $ which means that $ x $ is a set of all those $ a $ for which $ P(a) $ holds. Does it mean that $ P(a) $ has to evaluate to true?
Then we have definition of intersection of $ x $ which looks like that $ \bigcap x = \{ a | \forall y (y \in x \rightarrow a \in y ) \} $. Let's assume that $ x = \{ \{ 1, 2 \}, \{ 1, 3 \} \} $ then $ \bigcap x $ should be equal to $ \{ 1 \} $ but I can also said that it is equal $ \{ 8 \} $, because $ y = \{ 8 \}$ is not an element of $ x $ so the $ P(a) $ is evaluated to true, so $ \bigcap x = \{ 8 \} $ which is wrong. Can you explain me what I am wrongly mixing? I know that normally I would assume that $ y \in x $ is true, but in this case the $ P(a) $ should be true, shouldn't it?