Solve these difficult and challenging questions!

1. Describe any statement that is the equivalent of (p q) given the base statement to be represented by (pq)(pq).

2. Determine the truth value of a quantified statement xP(x) with real numbers constituted in the domain where P[x] is a representation of "x = x + 7."

3. Describe the denotation of the statement x y ((x < 0) (y < 0) (xy > 0)).

4. Determine the truth value of the statement n(n + 1 > n) given that the domain constitutes of all the real numbers.

5. Given the denotation of P[x[ as "X > 7", determine the value of x given that the statement is true.

6. Describe and expound on the statement that illustrates that zero is the output of a real number minus itself.

7. Determine the truth value of the quantified statement xQ(x) where the real numbers are the domains and Q[x] represents the statement "x < 5."

8. Describe with a logical explanation the statement equivalent to p q.

9. Given the quantified statement xR(x), determine the truth value where R[x] represents the statement "x > 2" and the domain is given as real numbers.

10. Counter illustrate the statement xy(xy > y) given that domain for the variables that consists of integers.