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In my Discrete Mathematics class I have just started learning about the laws of propositional logic. I've got about halfway through a problem but am now stuck for a while now and I don't understand where to go/what law to use. Any guidance would be appreciated!

Problem: ( → ¬( ∨ )) ∧ (( → ) ∧ ( ∧ ))

My work so far: (p→(¬p ∧ ¬q)) ∧ ((¬r ∨ p)∧(p ∧ q))

            (¬p ∨ ¬p ∧ ¬q) ∧ ((p ∨ ¬r) ∧(p ∧ q))

                 (¬p∧ ¬q)  ∧  *Stuck here: Only thing I can think is distributive law but 
                               signs aren't right
Schoggi_23
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1 Answers1

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Please note the $\lnot p \lor(\lnot p\land \lnot q)$ in not equivalent to $\lnot p\land\lnot q$, but to $\lnot p$. This follows from the absorption law $a\lor (a\land b) = a$.

Now, for your question: Note that $\land$ is assoctiative. Thus $$ (p\lor\lnot r)\land(p\land q) = ((p\lor\lnot r) \land p) \land q $$

But I do not advice you to use distributivity there. Rather, since the first term is $\lnot p$ try to write it like this $$ (\lnot p) \land ((r\to p) \land p \land q) = (\lnot p) \land (r\to p) \land p \land q $$ and you might see something.

Lazy
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