consider the statement (formula) (3x)A(x) → A(z) where z is a new variable not free (not an "input variable") in A(x). Find now a specific example of A(z) over the set N and choose a specific value of z EN so that (1) becomes false (meaning we cannot prove it, since proofs start from true axioms and preserve truth at every step). (1)

Let P(x) = (x - 1)(x - 2)²(x-3)³. Find the set of such that X (a) P(x) = 0, (b) P(x) 0.

40. Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. (a) Vxy(x = 1/y) (b) Vy(y²-x< 100) (c)(x²y³)

Additional exercises * EXERCISE 1.8.1: Proving statements about odd and even integers with direct proofs. Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k + 1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. Prove each of the following statements using a direct proof. (a) The sum of an odd and an even integer is odd. (b) The sum of two odd integers is an even integer. (c) The square of an odd integer is an odd integer. (d) The product of two odd integers is an odd integer. (e) If x is an even integer and y is an odd integer, then x2 + y? is odd. (f) If x is an even integer and y is an odd integer, then 3x + 2y is even. (g) If x is an even integer and y is an odd integer, then 2x + 3y is odd. (h) The negative of an odd integer is also odd. (1) If x is an even integer then (-1)* = 1 1) If x is an odd integer then (-1)* = -1 Feedback?