Question

Let Q(x) denote the statement "x + 1 = 2x". If the universe of discourse is all integers, what are these

truth values?

a) Q(0)

b) Q(-1)

c) Q(1)

d) existential quantifier xQ(x)

e) universal quantifier xQ(x)

f) ~universal quantifier xQ(x)

g) existential quantifier x~ Q(x)

Answer 1

a) Q(0) is false because $0+1 \neq 2 \cdot 0$;

b) Q(-1) is false because $-1+1 \neq 2 \cdot (-1)$;

c) Q(1) is true because $1+1 = 2 \cdot 1$;

d) $\exists xQ(x)$ is true because there exists x (x=1) that makes statement Q(x=1) to be true;

e) $\forall xQ(x)$ is false because Q(x) is not true for all x, see a) and b)

f) $\lnot \forall xQ(x) = \exist x \lnot Q(x)$ is true because there exists such a x for which Q(x) is not true, e.g. a) and b)

g) $\exist x \lnot Q(x)$ is true the same as f)