Answer
See explanation
Work Step by Step
a) \begin{align*}
P \downarrow P &\equiv\sim(P\vee P)&&\text{by definition of } \downarrow\\
&\equiv \sim P && \text{by idempotent law over } \vee
\end{align*}
b) \begin{align*}
(P\downarrow Q)\downarrow(Q\downarrow P)&\equiv\sim(P\vee Q)\downarrow\sim(Q\vee P)&&\text{by definition of } \downarrow\\
&\equiv \sim(P\vee Q)\downarrow\sim(P\vee Q)\equiv\sim[\sim(P\vee Q)] && \text{by idempotent law over } \\
&\equiv P\vee Q
\end{align*}
c) $(P\downarrow P)\downarrow(Q\downarrow Q)\equiv\sim P\downarrow \sim Q\equiv\sim(\sim P\vee\sim Q)\equiv\sim(\sim P)\land\sim(\sim Q)\equiv P\land Q$
d) $P\rightarrow Q\equiv\sim P\vee Q\equiv(P\downarrow P)\vee Q\equiv\sim[\sim((P\downarrow P)\vee Q)] \equiv \sim[(P\downarrow P) \downarrow Q] \equiv [(P\downarrow P) \downarrow Q] \downarrow [(P\downarrow P) \downarrow Q]$
e) $P\leftrightarrow Q\equiv (P\rightarrow Q)\land(Q\rightarrow P)\equiv [(P\rightarrow Q)\downarrow (P\rightarrow Q)]\downarrow [(Q\rightarrow P) \downarrow (Q\rightarrow P)] \equiv [[(P\downarrow P)\downarrow Q] \downarrow [(P\downarrow P) \downarrow Q] \downarrow [(P\downarrow P) \downarrow Q] \downarrow [(P\downarrow P) \downarrow Q]] \downarrow [[(Q\downarrow Q)\downarrow P] \downarrow [(Q\downarrow Q) \downarrow P] \downarrow [(Q\downarrow Q) \downarrow P] \downarrow [(Q\downarrow Q) \downarrow P]]$
