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I need help showing that $[ (p \land q) \Rightarrow (p \Rightarrow q) ]$ is a tautology by applying a chain of logical identities. The question also asks to identify each identity I use. I have no clue where to start.

Jean Marie
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Spencer Jarocki
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  • Please check here for MathJax tutorial – IamKnull Sep 25 '19 at 03:44
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    What logical identities are available to you to use? And my usual approach to such problems is to figure out why it's true in the first place. Once I understand that, it becomes much easier to apply the axioms that result in a formal proof. – Robert Shore Sep 25 '19 at 03:52

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You have to know that $(p \implies q) \iff (\neg p \lor q)$ (This can be proven by truth tables). Then, you begin start with $p \land q$, if that is true, then, $p$ and $q$ have to be true. $(p \land q) \implies q$

if $q$ is true, then, it doesn't matter if we ask if $q \lor r$ is true, because, we know it is, since $q$ is true. $q \implies (q \lor \neg p)$.

Since $\lor$ operator is conmutative $(q\lor \neg p) \iff (\neg p \lor q)$. Then, we have

$(p \land q) \implies q \implies (q \lor \neg p) \implies (\neg p \lor q)$

By transition $p\land q \implies \neg p \lor q$

Jonatan Garcia
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