Discrete mathematics
Show that the proposition ~(p^q) and ~pv~q are logically equivalent.
To show that ~(p^q) and ~p v ~q are logically equivalent, we need to show that they have the same truth value for all possible truth values of p and q.
First, let's consider ~(p^q). The negation of p ^ q is equivalent to ~(p∧q) = ~p v ~q. This is known as De Morgan's law for propositional logic.
So, ~(p^q) = ~p v ~q
Now, let's examine ~p v ~q. This proposition is true if either ~p is true or ~q is true (or both). This is also true if p and q are false. Therefore, ~p v ~q is equivalent to ~(p^q).
Therefore, ~(p^q) and ~p v ~q are logically equivalent.