Solution to (a) Let P(x) be the statement x 2 ≥ x. (i) What are truth values … - Sikademy
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Archangel Macsika

(a) Let P(x) be the statement x 2 ≥ x. (i) What are truth values of the propositions P(1) , P(-1) , P(0) and P( 1 2 )? (ii) What is the truth value of the proposition ∀xP(x), where the domain consists of all real numbers? (iii) What is the truth value of the proposition ∀xP(x), where the domain consists of all integers? (b) Let Q(x,y,z) be the statement x 2 + y 2 = z 2 . What are the truth values of (i) ∀(x,y,z) Q(x,y,z) (ii) ∃(x,y,z) Q(x,y,z) Where the domain consists of all real numbers.

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a) P(x):x^2\geq x

i)

P(1): 1=1 - true

P(-1): 1>-1 - true

P(0): 0=0 - true

P(1/2): 1/4<1/2 - false


ii)

Answer: false (example: P(1/2): 1/4<1/2 )


iii)

Answer: true


b) Q(x,y,z):x^2+y^2=z^2

i)

Answer: false


ii)

Answer: true (example: 3^2+4^2=5^2 )

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