$P$: $((p \land q) \vee r)\implies (l \vee t)$... If $p$,$q$,and $l$ are all false and $r$ and $t$ are true determine if $P$ is true

Question

$P$: $((p \land q) \vee r)\implies (l \vee t)$... If $p$,$q$,and $l$ are all false and $r$ and $t$ are true determine if $P$ is true.

How would I show this just by logically writing out $p$ and $q$ false or $r$ true implies $l$ false or $t$ true so $P$ is true? Or would I have to make a truth table which would be complicated. $P$ is true right?

Answer 1

Doing a truth table is one way to prove it. If you don't want to do it you can argue as follows.

Whenever the consequent of a conditional statement is true, the conditional statement is true. Since $t$ is true, so is $l\lor t$, therefore $P$ is true.

Answer 2

Note this: $ ((p \land q) \vee r) \rightarrow ((F \land F) \vee T) \rightarrow F\vee T\equiv V\\ (l\vee t) \rightarrow (F \vee T) \equiv T\\ T \Rightarrow T \equiv T $

Then $P$ is true.