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$P$: $((p \land q) \vee r)\implies (l \vee t)$... 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?

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  • $\begingroup$ The title isn't supposed to be the first line of your question. $\endgroup$ – Git Gud Apr 30 '14 at 23:37
  • $\begingroup$ Its not a title, if I just have the second part you would not know what the problem was and visa versa. $\endgroup$ – user146167 Apr 30 '14 at 23:40
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    $\begingroup$ Exactly. I just look at the question (I just don't read "titles" once I open the question) and I can't make head or tails with what it is. It doesn't make sense. $\endgroup$ – Git Gud Apr 30 '14 at 23:42
  • $\begingroup$ Well I don't know what type of problem it is to call it anything different, and the way I look at filling out the boxes is question on the top box and then any addition thoughts I have on solving it or questions i have about it. $\endgroup$ – user146167 Apr 30 '14 at 23:51
  • $\begingroup$ In fact when you go to post a question, the spot for title says what is the question? $\endgroup$ – user146167 Apr 30 '14 at 23:55
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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.

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  • $\begingroup$ I would have to use that same logic for the other side where p and q false or r true also right? $\endgroup$ – user146167 May 1 '14 at 0:12
  • $\begingroup$ The LHS of $\implies$ doesn't matter, that's the whole point of the remark "Whenever the consequent of a conditional statement is true, the conditional statement is true". A conditional statement is false only if the antecedent is true and the consequent is false. So if the consequent is true, necessarily the conditional statement is true. Is it clearer? $\endgroup$ – Git Gud May 1 '14 at 0:13
  • $\begingroup$ Ok so say the whole thing was $ a \implies b$ the only time that the proposition is false is where a is true and b is false, i get that but see where a and be and both false the proposition is true ...ahhh i got it you showed that b was true and in both cases of a the proposition was true ....Thanks :) $\endgroup$ – user146167 May 1 '14 at 0:23
  • $\begingroup$ @user146167 Yes, it doesn't matter what $a$ is. No problem. $\endgroup$ – Git Gud May 1 '14 at 0:24
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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.

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