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
This will be proven by contradiction.

Let such an operation $\circ$ exist.

Truth table of $\land$:

From the truth table, we can see that $q=F \implies \forall P \in \left\{{T,F}\right\}: P \land Q=F$.

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 $\circ$.