Simplify the proposition ¬(p∧q)∧(p∨¬r)

Question

I can not come up with anything concrete,

$$¬(p∧q)∧(p∨¬r)$$

Thanks!

Answer 1

Nnn...nope, that's about as simple as you will get.

Use deMorgan's Negation and you obtain CNF (Conjunctive Normal Form).

$$\lnot(p\land q)\land(p\lor\lnot r) ~\iff~ (\lnot p\lor \lnot q)\land (p\lor\neg r)$$

Answer 2

According to De Morgan's Law,

$\neg(p\cap{q})\cap(p\cup\neg{r})\equiv(\neg{p}\cup\neg{q})\cap(p\Rightarrow{r})$. Look up for the Truth table, that will give you a better clue on how to approach such questions.

Answer 3

Any solution of the form $p \circ q \circ' r$ needs parenthesis, and this means that we are effectively looking at $p \circ'' s$ which isn't complex enough to return the values required.