$($P $->$ Q$)($P $$ Q$)$

$1.$ Grow the truth tree for the proposition. Be sure you check off a formula after you decompose it$.$

If a path is a closed path, put a cross at the end of the path. Please note a fully grown tree may contain ONLY checked formulas and atomic formulas.

$2.$ Based on your truth tree, are $($P $->$ Q$)$ and $($P v Q$)$ logically equivalent? Why or why not?

Please explain your reasoning.

$3.$ What justifies your conclusion? An example, a counter example, or the whole truth tree? In case

of an example or a counter example, please provide it$.$