Propositional logic simplification exercise

Question

I am trying to simplify this to use just one of $\neg,\wedge,\vee,\leftrightarrow,\rightarrow$ using proportional laws $\neg\,[\neg\,(p∧r)∨(\neg\,p∧q)]∧(p∨r)$ but I have no luck.

Answer 1

You need to know: $\neg(a\wedge b)=\neg a\vee\neg b$ and $\neg(a\vee b)=\neg a\wedge\neg b$.

So you start by simplifying to $$(p\wedge r)\wedge\neg(\neg p\wedge q)\wedge(p\vee r)$$ then to $$(p\wedge r)\wedge(p\vee\neg q)\wedge(p\vee r)$$ But $p\wedge r$ implies both $p\vee\neg q$ and $p\vee r$, so the expression reduces to $$p\wedge r$$