Problem 7 - Predicate Logic

For the following interpretation ( $D = \{ a, b\}$), $P(a, a)=T$, $P(a, b)=F$, $P(b, a)=F$, $P(b, b)=T$, determine the truth value of the following formulas.

1.

$(\forall x) (\exists y) P(x,y)$;

2.

$(\exists x) (\forall y) P(x,y)$;

3.

$(\exists y) \neg P(a, y)$;

4.

$(\forall x)(\forall y)(P(x,y) \rightarrow P(y,x))$.

$P(a, a)=T$ and $P(b, b)=T$, therefore $(\forall x) (\exists y) P(x,y) = T$.

$P(a, a)=T$, but $P(a, b)=F$ and $P(b, b)=T$, but $P(b, b)=F$, therefore $(\exists x) (\forall y) P(x,y) = F$.

$P(a, b)=F$, therefore $\neg P(a, b) = T$, therefore $(\exists y) \neg P(a, y) = T$

$P(a, b) \rightarrow P(b, a) = F$, therefore $(\forall x)(\forall y)(P(x,y) \rightarrow P(y,x)) = F$.