Problem 6 - Logic

Consider the following formula:

$$A: (\exists x) P(x) \rightarrow (\forall x) P(x)$$

1.

Prove this formula is always true if the domaine $D$ contains onlye one element.

2.

Let $D = {a, b}$. Find an interpretation over $D$ in which $A$ is evaluated to false.

$A \rightarrow B$, $\neg A \vee B$
T	T	T
T	F	F
F	F	T
F	T	T

$$A: [\neg (\exists x) P(x)] \vee [(\forall x) P(x)]$$

$$A: [(\forall x) \neg P(x)] \vee [(\forall x) P(x)]$$

If $D = {a}$, then $A: \neg P(a) \vee P(a)$, which is a tautology.

If $D = {a, b}$ with $P(a)$ and $\neg P(b)$, then $A$ is a contradiction.