We illustrate why the negation of 
is
using two sets X and Y,
each having two elements. In the diagrams below, if p(x,y) is true we
mark the square red and if it is false then we leave it white. There are
of course 16 such diagrams.
Now consider the statement . For
this to be true we must have one of the following diagrams. This is because
for at least one fixed x and every element of Y the statement is true.
Thus there must be at least one red row.
It is false for the remaining nine diagrams shown below (since no row is all red).
The situation is reversed for the statement
. In this case, the statement
is true if there is no completely red row and false when there is.