The statement I am trying to simplify is: $\lnot(p\lor\lnot q)\to(p\to(p\land\lnot p))$

First thing I did was use the Material Implication Law resulting:
$\lnot\lnot(p\lor\lnot q)\lor(\lnot p\lor(p\land\lnot p))$
$(p\lor\lnot q)\lor(\lnot p\lor(p\land\lnot p))$

Then I distributed the right side resulting in:
$(p\lor\lnot q)\lor((\lnot p\lor p)\land(\lnot p\lor\lnot p))$

And since $(\lnot p\lor\lnot p)=\lnot p$:
$(p\lor\lnot q)\lor((\lnot p\lor p)\land\lnot p)$

And since $\lnot p\lor p=\top$:
$(p\lor\lnot q)\lor(\top\land\lnot p)$

Up to there is where I feel stucked and I am not sure if I did it correctly and would like appreciate some feedback if its correct or wrong and what can I do to solve it better. Thanks in advance!

  • $\begingroup$ I'm no expert, so I'll ask a perhaps silly question: are you trying to find the most simplified form of the expression, or are you trying to find what it spits out (i.e. T/F) given different values of $p$,$q$ being T/F? $\endgroup$
    – scoopfaze
    Jun 26, 2019 at 18:26
  • $\begingroup$ Haha it's okay, I am simply just looking for the most simplified form of the expression. $\endgroup$ Jun 26, 2019 at 18:27
  • 1
    $\begingroup$ Hint: $\top\land \phi\equiv\phi$ for any formula $\phi$ $\endgroup$
    – Vsotvep
    Jun 26, 2019 at 18:31

1 Answer 1


Given expression $$ \neg (p \vee \neg q) \rightarrow ( p \rightarrow (p \wedge \neg p))$$ $$ = \neg (p \vee \neg q) \rightarrow ( \neg p \vee F) $$ $$ = ( p \vee \neg q) \vee \neg p = (T \vee \neg q) = T$$ enter image description here As you are stuck after this: $$( p \vee \neg q) \vee ( T \wedge \neg p)$$ On using $ T \wedge \neg p = \neg p$, the statement becomes, $$ p \vee \neg q \vee \neg p$$ On using $\neg p \vee p = T$ , the statement becomes, $$ T \vee \neg q = T$$

  • $\begingroup$ $\top\vee \varphi \equiv \top$ for any $\varphi$ $\endgroup$ Jun 27, 2019 at 1:02
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    $\begingroup$ @Graham thank you! I am used to work with +,• and ‘, this notation confuses me slightly. $\endgroup$
    – xrfxlp
    Jun 27, 2019 at 2:11

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