For the universe of all integers, let p(x), q(x), r(x), s(x), and r(x) be the following open statements.
p(x): x > 0
q(x): x is even
r (x): x is a perfect square
s(x): x is (exactly) divisible by 4
f (x) : x is (exactly) divisible by 5
(a) Write the following statements in symbolic form.
(i) At least one integer is even.
(ii) There exists a positive integer that is even.
(iii) If x is even, then x is not divisible by 5.
(iv) No even integer is divisible by 5.
(v) There exists an even integer divisible by 5.
(vi) If x is even and x is a perfect square, then x is divisible by 4.
(b) Determine whether each of the six statements in part (a) is true or false. For each false statement, provide a counterexample.
(c) Express each of the following symbolic representations in words.
(i) ∀x [r(x) → p(x)]
(ii) ∀x [s(x) → q(x)]
(iii) ∀x [s(x) → ¬t(x)]
(iv) ∃x [s(x) ∧ ¬r(x)]
(d) Provide a counterexample for each false statement in part (c).