(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)
i)
- true
- true
- true
- false
ii)
Answer: false (example: )
iii)
Answer: true
b)
i)
Answer: false
ii)
Answer: true (example: )