Solution to 2. Let P(x) and Q(x) be the statements x3 < 50 and x+2 < 5 … - Sikademy
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Archangel Macsika

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))

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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 )

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