Answer to Question #122161 in Discrete Mathematics for Kavee

Question

2. Let P(x) and Q(x) be the statements x3 < 50 and x+2 < 5 respectively. Find the truth values of following quantifications, where the domain consists of all real numbers.
(i) ∃x P(x)
(ii) ∀x P(x)
(iii) ∀x Q(x)
(iv) ∃x Q(x)
(v) ∀x (¬P(x))
(vi) ∃x (¬Q(x))
(vii) ∃x> 0 (P(x))
(viii) ∀x> 0 (¬Q(x))
(ix) ∀x (P(x) ∨ Q(x))
(x) ∃x (P(x) ∧¬Q(x))

Answer 1

i) truth (ex.: $2^3<50$ )

ii) false (ex.: $4^3>50$ )

iii) false (ex.: $8+2>5$ )

iv) truth (ex.: $1+2<5$ )

v) $\neg P(x):\{x^3\geq50\}$

Answer: false (ex.: $2^3<50$ )

vi) $\neg Q(x):\{x+2\geq5\}$

Answer: truth (ex.: $3+2=5$ )

vii) truth (ex.: $2^3<50$ )

viii) false (ex.: $1+2<5$ )

ix) $P(x)\lor Q(x):\{x^3<50$ or $x+2<5\}$

Answer: false ( ex.: $4^3>50$ and $4+2>5$ )

x) $P(x)\land\neg Q(x): \{x^3<50$ and $x+2\geq5\}$

Answer: truth (ex.: $3^3<50$ and $3+2=5$ )