How to validate below proposition ?

Question

(p->q) ^ (r->s)

~q OR ~s

 

Conclusion is ~p OR ~r

 

How to see if the conclusion is valid or not ?

Answer 1

here ~q OR ~s can be written as q->~s.

so equation becomes (p->q) ^ (r->s) ^ (q->~s) 

(p->q) ^ (q->~s) = (p->~s) (transitive rule)

(p->~s) ^ (r->s) = (p->~s) ^ (~s->~r) (taking contrapositive)

                         = (p->~r) (transitive rule)

                         =(~p v ~r)

                        = ~p or ~r , HENCE PROVED.