1. Show that the proposition ~(p^q)and ~pv~q are logically equivalent.
2. Show that
a. ~(pvq)v(~p^q) equivalent (~p^~q)v(~p^q) "De Morgan's law"
b. ~(pvq)v(~p^q)v(~p^q) equivalent ~p^(~q^p)"Distributive law"
1. To show that ~(p^q) and ~pv~q are logically equivalent, we can use De Morgan's law:
~(p^q) = ~p v ~q
~pv~q = ~(p^q)
Therefore, ~(p^q) and ~pv~q are logically equivalent.
2.
a. ~(pvq) v (~p^q) equivalent to (~p^~q) v (~p^q) (De Morgan's law)
Distributing the negation over the conjunction in both expressions:
~p v ~q v ~p ^ q v ~p ^ ~q
= ~p ^ ~q v ~p ^ q (commutativity of disjunction)
= ~p ^ (~q v q) (distributive law)
= ~p ^ ⊥ (contradiction)
= ~p (identity law)
Therefore, ~(pvq) v (~p^q) is equivalent to ~p.
b. ~(pvq) v (~p^q) v (~p^q) equivalent to ~p ^ (~q^p) (Distributive law)
Distributing the negation over the disjunction in the expression:
~p ^ ~q v q v ~p ^ q v ~p ^ q
= ~p ^ ~q v ~p ^ q (idempotent law)
= ~p ^ (q v ~q) (commutativity of conjunction)
= ~p ^ ⊤ (tautology)
= ~p (identity law)
Therefore, ~(pvq) v (~p^q) v (~p^q) is equivalent to ~p^(~q^p).