Question

Let Q(x) be the statement “x + 1 > 2x.” If the domain consists of all integers, what are these truth values? (7 pts.)

a) 𝑄(0) b) 𝑄(−1) c) 𝑄(1)

d) ∃𝑥𝑄(𝑥) e) ∀𝑥𝑄(𝑥) f) ∃𝑥¬𝑄(𝑥) g) ∀𝑥¬𝑄(𝑥)

Answer 1

Let $Q(x)$ be the statement “$x + 1 > 2x$ ”. If the domain consists of all integers, let us find the following truth values.

a) Since $1>0,$ we get that $𝑄(0)$ is true.

b) Taking into account that it is not true that $0>-2,$ we get that $𝑄(-1)$ is false.

c) Taking into account that it is not true that $2>2,$ we get that $𝑄(1)$ is false.

d) Since $1>0,$ we get that $𝑄(0)$ is true, and hence $∃𝑥𝑄(𝑥)$ is true.

e) Taking into account that it is not true that $0>-2,$ we get that $𝑄(-1)$ is false, and hence $∀𝑥𝑄(𝑥)$ is false.

f) Since it is not true that $2>2,$ we get that $𝑄(1)$ is false. Therefore, $\neg 𝑄(1)$ is true, and thus $∃𝑥¬𝑄(𝑥)$ is true.

g) Since $1>0,$ we get that $𝑄(0)$ is true. We conclude that $\neg 𝑄(0)$ is false, and hence $∀𝑥¬𝑄(𝑥)$ is false.