Proving Logic statement

Question

So I have an statement that I need to prove using Logical Equivalences: $$(p\land q) \lor [p \land (\lnot( \lnot p \lor q)) ] \equiv p $$ I made it through some steps but I can't seem to make it to the end. Here is my work: $$ \equiv (p\land q) \lor [p \land ( p \land \lnot q)) ] $$ $$ \equiv (p\land q) \lor [(p \land p) \land \lnot q] $$ $$ \equiv (p\land q) \lor (p \land \lnot q) $$ This is about as far as a I get. Can someone show me where I went wrong or point me in the right direction?

Answer 1

You’re almost there: $(p\land q)\lor(p\land\neg q)\equiv p\land(q\lor\neg q)$ by one of the distributive laws. Can you finish it now?