is called the image of the set and
is called the pre-image of and all the facts in Plato's PDF document are proven with these definitions. Pinter turns these into functions on the power sets. I wonder why? What use is made of image and pre-image as functions? Do we get anything extra out of this idea? (I guess this could be just a teaching device to drive home the fact that the inputs and outputs to the image and pre-image are sets. Or maybe it's a better notation.)
So I went back to my old texts and found the most precise one, the reference text, does define the image and pre-image as maps between the power sets. It calls these "the induced maps." But it doesn't introduce any new notation or prove anything different than the others. So I conclude this is the most precise definition, but nothing very substantial comes out of it.