Set Theory Notation Crises

For those who are familiar with the following notation, could you explain it in plain English because I picked up a set theory textbook but the book assumes the reader is familiar with the notation without giving a formal explanation anywhere.

1) $$\{x:\mathscr{P}x\}$$ 2) $$\{\mathscr{a}_x: \mathscr{P}x\}$$ 3) $$\left(\bigcup_{i \in I} \mathscr{a}_i\right)^{c}$$
4) $$x \in \{y: y>2\}$$ 5) (distributive laws) $$B \cap \left(\bigcup_{i \in I} \mathscr{a}_i\right)$$ 6) $$\{x: x \not\in x\}$$

The Px is like a P that is curly and looks like old english writing or might be a greek and same with a_i. a is like a fat curly a which I'm thinking might stand for "for all." Any input with explanation would be great or any reference would work as well. The book I'm reading is 'Set Theory An Introduction' by Robert Vaught.

• Do you mean like this? – Stahl May 19 '13 at 6:15
• It's really hard to imagine that an introductory book wouldn't explain its notation anywhere. It doesn't sound like a book from which you'll be able to learn very much. – Gerry Myerson May 19 '13 at 6:19
• @Stahl Wow, that's pretty close. The only mistake is that for a, it is in lower case but might be the same thing. Also for 3), it is like a curly brace but I am almost positive its a complement sign. Besides that, you hit it spot on. – user67527 May 19 '13 at 6:21
• @GerryMyerson Maybe I overexaggerated when I posted the question but it does have some explanation but I'm not quite understanding it which is making it a bit challenging since this is my first time ever touching set theory. – user67527 May 19 '13 at 6:24
• I'm not sure what the "curly brace" would be, but I think that should do it for the a's and the rest. (Possibly they're bigger and in a different font, but I think this should get the idea across. I assume they're arbitrary sets anyway, in which case it won't really matter what symbol you use to denote them.) – Stahl May 19 '13 at 6:26

1) and 2) are explained in page 8. ${\scr P} x$ indicates that $x$ satisfies some predicate or some property (for example consider $x$ satisfies $\scr P$ if and only if $x$ is a positive integer then $2,23$ does not satisfy $\scr P$ and therefore is not an element of $\{x:{\scr P} x\}$), and $a _x$ is a function of $x$, so you can just substitute that by $f(x)$ for some fixed $f$ defined whenever $x$ satisfies $\scr P$.
6) Is an instance of 1) it is therefore a set formed out of all the sets $x$ so that $x$ is not a member of itself, it is generally used to show a paradox which arises when one allows himself to form sets $\{x:{\scr P} x\}$ without any restriction on the predicate $\scr P$.
$\bigcup_{i\in I} a_i$ Is the set formed by all elements $x$ such that $x\in a_i$ for at least one $i\in I$, here the $a_i$ are said to form a family of sets indexed on the set $I$, so 5), becomes just the intersection of this family with $B$, that is, elements that lie both in $B$ and in some $a_i$, the (distributive law) refers to the property $$B\cap (\bigcup_{i\in I} a_i) = \bigcup_{i\in I} (B\cap a_i)$$ Now given a set $a$, $a^c$ usualy refers to the complement of the set (which is seen to be contained in some ommited larger set), thus 3) is defined if all of the $a_i$ lie inside the same set $A$ in which case it expands to $$(\bigcup_{i\in I}a_i)^c = A\setminus (\bigcup_{i\in I} a_i)$$
4) Is, again, an instance of 1) where by saying that $x\in \{y:y>2\}$ we mean $x>2$ (note that in order for this to make sense $x$ must be a real number)