Conjunction has no Inverse

Theorem
Let $\land$ denote the conjunction operation of propositional logic.

Then there exists no binary logical connective $\circ$ such that:


 * $(1): \quad \forall p, q \in \left\{{T, F}\right\}: \left({p \land q}\right) \circ q = p$

Proof
Let $q$ be false.

Then $p \land q = F$, whatever truth value $p$ holds.

Either $F \circ F = T$ or $F \circ F = F$, but not both.

So if $q = F$ either:
 * $\left({p \land q}\right) \circ q = T$

or:
 * $\left({p \land q}\right) \circ q = F$

If the first, then $(1)$ does not hold when $p = F$.

If the second, then $(1)$ does not hold when $p = T$.

Hence there can be no such $\circ$.