For the universe of all integers, let p(x), q(x), r(x),s(x), and t (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
t(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 xis 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 xis 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).
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