Show that
a. ~(pvq)v(~p^q) equivalent (~p^~q)v(~p^q)"De Morgan's law
b. ~(pvq)v(~p^q) equivalent ~p^(~pvq) "Distributive law"
To prove the equivalence of the given expressions, we need to show that they represent the same truth values for all possible truth values of p and q.
a. De Morgan's law:
~(p ∨ q) ∨ (~p ∧ q) equivalent to (~p ∧ ~q) ∨ (~p ∧ q)
To show this, let's consider the truth table for both expressions:
| p | q | ~(p ∨ q) ∨ (~p ∧ q) | (~p ∧ ~q) ∨ (~p ∧ q) |
|---|---|----------------------|-----------------------|
| T | T | F | F |
| T | F | F | F |
| F | T | F | F |
| F | F | T | T |
As we can see from the truth table, the truth values for both expressions are the same for all possible truth values of p and q.
b. Distributive law:
~(p ∨ q) ∨ (~p ∧ q) equivalent to ~p ∧ (~p ∨ ~q)
Following a similar process as in part a, we construct the truth table for both expressions:
| p | q | ~(p ∨ q) ∨ (~p ∧ q) | ~p ∧ (~p ∨ ~q) |
|---|---|----------------------|----------------|
| T | T | F | F |
| T | F | F | F |
| F | T | F | F |
| F | F | T | T |
The truth values for both expressions are the same for all possible truth values of p and q, confirming the equivalence of the two expressions.