Help with discrete Mathematics

[catala]

Hi!

I have an argument and I have to prove the validity of all possible ways.

I have proved by logical implication, tautology, contradiction and contrapositive, but the problem is reduced to prove the hypothesis by logical equivalences and implications.

The reasoning is as follows:

[P->(L->M)]^L^M -> ¬P

I've tried to do the following:

[P-> (L-> M)] ^ L ^ M

<=> [(P ^ L) -> M] ^ L ^ M {Export}

<=> (¬ ¬ P v L v M) ^ L ^ M {Involvement and Morgan}


From here not continue, and I've tried other ways and always came to the same and not continue.

Could anyone help me out?

Thanks! ;)

 

[Valkarie]

You can use the De Morgan's law to simplify the expression. [P-> (L-> M)] ^ L ^ M<=> [(P ^ L) -> M] ^ L ^ M {Export}<=> (¬(¬P ^ ¬L) v M) ^ L ^ M {Involvement and Morgan}<=> [¬¬P v ¬¬L v M] ^ L ^ M {Double Negation}<=> [P v L v M] ^ L ^ M {De Morgan's Law}<=> (P v M) ^ L {Distributive Law}<=> (P v M) ^ (L ^ L) {Identity Law}<=> (P v M) ^ L {Idempotent Law}<=> ¬(P v M) v L {Implication}<=> ¬P ^ ¬M v L {De Morgan's Law}<=> ¬P v L {Distributive Law}<=> ¬P -> L {Implication}