To put the joke in straight english for rosala and melody (because my current state is called evasive study behaviour).
If Jack is one of all the boys in the world and beth is one of all the girls in the world.
Then (it assumes) there must be some function linking all the boys and girls in the world with their perfect date. So If you enter eg. Rosala you get the perfect date for Rosala.
Now we are looking for the perfect date of Beth, so we enter Beth into the function.
Beth knows from previous experiences that there's a certain type of boy Beth likes to take on a date.
This certain type, which is (off course) a part of all the boys in the world, is a good dancer, a good conversationalist, a gentleman and (obviously) free.
She then states that it was previously determined that Jack is a good dancer, a good conversationalist and a gentlemen.
Since she obviously doesn't know whether Jack is free, she states this is to be determined by the reader.
(which is a joke, since in actual proofs, this is merely a statement saying if the reader is interested he should find out how it works himself)
Since Jack has all the parts of the date Beth wants, he's a solution to the 'perfect date' function.
Now that was a lesson on set notation