$(20$ points$)$ Propositional Operations and Equivalences

All propositional logic statements can be expressed by using not,${?}^{{?}^{?}},vv,,$harr,$T($true$)$ and $F($false$).$ Not all

of these $7$ symbols are vital to express all statements. For instance instead of $F$ we can use notT, or instead

of $P^{{?}^{?}}Q$ we can use not$(notPvvnotQ).$ Actually PharrQ is usually interpreted as $(PQ{)}^{{?}^{?}}(QP)$ most of

the time. So if we have $$ and ${?}^{{?}^{?}}$ we dont need harr at all. That is to say a shorter list of symbols might be

enough to express all possible statements.

Show that $$ and $F$ is enough to express any propositional logic statement. $($Express notP,$P^{{?}^{?}}Q,PvvQ$

and $T,$ by using only $P,Q,$ and $F.)$