# I need assistance trying to simplify a logic statement

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!

• 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? Jun 26, 2019 at 18:26
• Haha it's okay, I am simply just looking for the most simplified form of the expression. Jun 26, 2019 at 18:27
• Hint: $\top\land \phi\equiv\phi$ for any formula $\phi$ Jun 26, 2019 at 18:31

Given expression $$\neg (p \vee \neg q) \rightarrow ( p \rightarrow (p \wedge \neg p))$$ $$= \neg (p \vee \neg q) \rightarrow ( \neg p \vee F)$$ $$= ( p \vee \neg q) \vee \neg p = (T \vee \neg q) = T$$ 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, $$T \vee \neg q = T$$
• $\top\vee \varphi \equiv \top$ for any $\varphi$ Jun 27, 2019 at 1:02