Let p(x), q(x),and r(x)be the following open statements.
p(x):x2 – 7x+10 = 0
q(x): x2 – 2x – 3 = 0
r(x):x < 0
a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation.
i) ∀x [p(x)→ ¬r(x)]
ii) ∀x [q(x) → r(x)]
iii) ∃x[q(x) → r (x)]
iv) ∃x[p(x) → r(x)]
b) Find the answers to part (a) when the universe consists of all positive integers.
c) Find the answers to part (a) when the universe contains only the integers 2 and 5.
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