How to formulate that there is no politician who is not ambitious in predicate logic?

Question

I want to go back to the logic of predicates. I tried to translate the following hrase:

There is no politician who is not ambitious : $p\wedge \not a $, where $p$ is politician. Is that correct ? Or should I write $\not\exists p,p\wedge \not a $, where $p$ is politician and a is being ambitious.

Or should I write $\forall p, a(p)$ ?

Answer 1

It is predicate logic; thus, we need predicates : $P(x)$ for "... is a Politician" and $A(x)$ for "... is Ambitious".

The formula will be :

$\lnot \exists x \ (P(x) \land \lnot A(x))$.

The formula is equivalent to $\forall x \ (P(x) \to A(x))$ that reads :

"Every Politician is Ambitious".