7. Let p(x), q(x), r(x), and s(x) be the following statements where x is an integer. p(x): x>0 q(x): xiseven r(x) : x is a perfect square s(x) : x is divisible by 4 t(x) : x is divisible by 5 (a) Express each of the following symbolic representations in words. (i) ∀x [r(x) → p(x)] (ii) ∀x [s(x) → q(x)] (iv) ∃x[s(x)∧¬r(x)] (v) ∀x[¬r(x)∨¬q(x)∨s(x)] (iii) ∀x [r(x) → ¬t(x)] (b) Determine whether each of the statements in part (a) above is true or false and provide a counter example for each false statement. (c) 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 integer divisible by 5 (vi) If x is even and x is a perfect square, then x is divisible by 4
7. Let p(x), q(x), r(x), and s(x) be the following statements where x is an integer. p(x): x>0
q(x): xiseven
r(x) : x is a perfect square s(x) : x is divisible by 4 t(x) : x is divisible by 5
(a) Express each of the following symbolic representations in words.
(i) ∀x [r(x) → p(x)] (ii) ∀x [s(x) → q(x)]
(iv) ∃x[s(x)∧¬r(x)] (v) ∀x[¬r(x)∨¬q(x)∨s(x)]
(iii) ∀x [r(x) → ¬t(x)] (b) Determine whether each of the statements in part (a) above is true or false and provide a counter
example for each false statement.
(c) 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 integer divisible by 5
(vi) If x is even and x is a perfect square, then x is divisible by 4
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