Ex. 1
Let D = {0, 1, 2, 3, ...}, and P(x) be "x is a factor of 12".
Then the truth set P(x) is {1, 2, 3, 4, 6, 12}.

The statement "There exist a fire truck that is red." has the negation "No fire trucks are red."
Let our domain, D, be the set of all fire trucks, and let the predicate Q(x) represent "x is red." 

We have
$ x in D such that Q(x).
negating gives us
~($ x in D such that Q(x)) Û " x Î D, Ø Q(x)..

Ex. 2
Let P(x) be "x is a factor of 12," Q(x) be "x is a factor of 4," and R(x) be "x < 5 and x ¹ 3," and suppose the domain of x is Z⁺. Use Þ and Û to indicate true relationships among P(x), Q(x), and R(x).
1. Q(x) Þ P(x)
2. R(x) Û Q(x)
3. R(x) Þ P(x)