Consider the statement 
. If this statement is true then its
truth set is U. Its negation must be false and have truth set the empty
set. Alternatively, if the statement is false (i.e. its truth set is a proper
subset of U) then its negation has to be true. We claim that the negation
is the statement 
. Clearly this is the negation if the
original statement is true. If the original is false then there must have
been some x for which p(x) was false. We conclude that the statement
is equivalent to .
Similarly, 
is equivalent to 
.
For statements involving more than one variable we have a similar situation.
We give below a table of statements involving two quantifiers along with its
negation.
