How to derive a truth value from the following formula, one where the formula is T and one where it is F

Question

Hi Guys I am trying to derive a structure from the formula where truth value for one structure is F and for another is T. I haven't seen a formula like this before so I am slightly confused.

this is the formula

∀ X • p(X , a) ∧ p(a, X ) ∧ (∃ Y • p(X , Y ) ∧ p(Y , X ))

the small a inside the brackets is really throwing me off. What would be the best way to proceed with this?

Answer 1

Hint

A simple way to "falsify" the formula:

$∀x \ [p(x,a) ∧ p(a,x) ∧ ∃y \ (p(x,y) ∧ p(y,x))]$

is to consider the domain $\mathbb N$ of natural numbers and interpret the individual constant $a$ as denoting the number $0$.

Finally, we have to interpret the binary predicate $p$ with the "less-then" ($<$) relation.

Clearly, $x < 0$ is false for any $x \in \mathbb N$ and thus:

$\mathbb N \nvDash ∀x \ [p(x,a) ∧ p(a,x) ∧ ∃y \ (p(x,y) ∧ p(y,x))]$.