Please help me solve this in an easy way:
Write an English statement representing $$\forall x\exists y:\lnot P(x,y).$$ Define all your terms first. Define $P$ and $x$ and $y$ first.
Answer:
The answer then as I understanded will be : x is a student at harvard , y represents items on the cafeteria menu. P(x,y) means student x likes item y on the menu.So the statement will be : All students at Harvard don't like some items on the cafeteria menu.
Well, as the problem says, you need to start by defining $P,$ $x,$ and $y.$ Both $x$ and $y$ should be categories of things, and $P$ should be a statement about those two categories of things. Once that's done, translate the logical sentence to English directly.
Edit: If we let $x$ range over students at Harvard, $y$ over items on the menu at the cafeteria, and $P(x,y)$ represent the statement "$x$ likes $y$," as you've suggested, then you are very close, indeed. The one thing that you didn't take into account is the symbol $\neg,$ which represents the negation of the statement that follows. So, for example, $\neg P(x,y)$ would represent the statement "$x$ does not like $y$." How does that alter your answer?
For example, if $P(x,y)$ is the statement "$x$ likes $y$", where $x$ ranges over people and $y$ ranges over things, your English statement could be the first clause of a well-known (at least in North America) advertising slogan.
The second clause is $\lnot \exists x \lnot P(x,s)$, where $s$ is the brand being advertised.
