If someone eats an entire cake, is it correct to say that he ate just a piece of cake? Can a whole cake still be considered a piece of cake if consumed in one sitting?
A whole cake would not normally be considered a piece of cake, which has the implication that the cake was divided into slices. If, however, eating the whole cake was very easy (perhaps the cake was of excellent quality, or not very large), then eating it could perhaps be described as a "piece of cake" in the idiomatic sense. See here, meaning 9 ... http://dictionary.reference.com/browse/piece+of+cake
If you want to go by the dictionary definition, I think you will find agreement that piece is not the whole cake. In fact, thefreedictionary.com actually uses this phrase as an example of that definition of piece:
Aside from the "official" definition, intuitively I think most native English speakers would agree that "I ate a piece of cake" necessarily implies that you did not eat an entire cake, because of that additional information, "a piece of". Otherwise, you would have said "I ate a cake" or "I ate cake".
The line between piece and whole is not completely clear though; for example, if someone scoops a bit of frosting with their finger, and then you eat everything else, technically you have not eaten the whole cake — does that mean you have eaten a piece? Again, I think most people would say no, but at what point, then, does it make the transition?
I imagine if you took a large group of people, and showed people pictures of cake (a full cake, 90% of a cake, 75%, half a cake, down to a sliver of cake), you would have disagreement over when you could call it a piece of cake starting from 100% to 50%, but the smaller you got, the more you'd see it called a piece. By the time you got to 50% of the cake or less I would imagine that nearly everyone would be willing to call it a piece (but as far as I know, nobody has run this experiment).
I think this boils down to the mathematical distinction between subset (which can be the whole set) and proper subset (which must be smaller).
In common usage, the word "piece" as well as the word "subset" implies a non-empty, proper subset. People are surprised when this is not the case, even if the word is technically correct.
Confusingly enough, some items are called "piece" in common usage, even if they're not separated from a whole. For instance, "piece of candy", "piece of chocolate" (both individually produced not cut out).
As it happens, I ate a cake just last night. My wife called from work later.
"How many pieces did you have?" she asked.
"Just one," I said. I couldn't have said otherwise because the cake was in one piece when she left the house, and I didn't cut it into smaller pieces.
This is the kind of thing that sitcoms are made of. Most human languages, English included, are rife with ambiguity. If someone eats an entire cake and claims to have eaten just one piece, you're right to be surprised because you know that "a piece" doesn't normally refer to an entire cake. They know it, too. But you also know what game they're playing, and they know that you know. So there's not much use in arguing about whether it's technically correct to say that the portion consumed was a single piece.
A piece is usually defined as
So the short answer is No it's not a piece, not in the usual sense of the word anyway.
But it shouldn't matter. It's Cake! :D
Refer to dbkk's answer.
Your question basically boils down to whether or not the canonical meaning of piece implies a proper subset or simply a subset.
For the actual distinction between a proper subset and a subset refer http://en.wikipedia.org/wiki/Subset .
I assume in my answer that the day-to-day usage implies a proper subset.