Set Theory Notation Translation

[MikeMHz]

Let X be a set of people and friends(x) be the set of friends of any x ε X. Then

friends is a function X -> P(X). We can extend friends to a function P(X) -> P(X) by

letting friends(Y) = U_yεY friends(y), for any Y C X.

 

I have been asked to translate into English a number of statements written in set notation based on this introduction.

Can somebody talk me through exactly what is being said here?

 

Also, two of the questions use "." to separate terms. I know "Ǝ" means "there exists" and "∀" means "for all"... but what about "."?

Edited October 13, 2012 by MikeMHz