Question

(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.

Answer 1

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: $3^2+4^2=5^2$ )