Using only the basic rules prove the following instance of D

Using only the basic rules, prove the following instance of DeMorgan’s Laws: (¬A & ¬B), .. ¬(A B)

Solution

Let P = (A v B)\' and Q = A\' ^ B\'

Let x be an arbitrary element of P then x P x (A v B)\'

x (A v B)

x A and x B

x A\' and x B\'

x A\' ^ B\'

x Q

Again, let y be an arbitrary element of Q then y Q y A\' ^ B\'

y A\' and y B\'

y A and y B

y (A v B)

y (A v B)\'

y P

Now combine (i) and (ii) we get; P = Q i.e. (A v B)\' = A\' ^ B\'