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