In all the questions below, we consider generic propositions P and Q and R$.$

$1.$ Is it true that $($P$)$ is equivalent to P$?$ Explain your answer using a truth$-$table with three columns $($one column for P$,$ one column for $$P$,$ and one column for$($P$)).$

$2.$ Consider the following expression: $($P $^$P$)->$ Q$.$ Does its truth$-$value depend on the truth$-$value of P$?$ Why?

$3.$ Write down truth$-$tables for the following expressions:

$1.($P $^$ Q$)$

$2.($P V $$Q$)$

$3.($P V Q$)$

$4.$P $^$ Q