Question

State DeMorgan's Laws of Boolean Algebra and verify them using a truth table.

Answer 1

Demorgan’s Law: This is the most powerful law of Boolean algebra. This states that:

1) (X Y)'  = X '.Y'

2) (X.Y)' = X' + Y'

The truth table for the second theorem is:

X	Y	X.Y	(X.Y)	X'	Y'	X'+Y'
0	0	0	1	1	1	1
0	1	0	1	1	0	1
1	0	0	1	0	1	1
1	1	1	0	0	0	0

To prove algebraically, we know that,

X + X' = 1 and X.X' = 0

So, if (X + Y)' = X'.Y' then
(X + Y) + X'.Y '= 1

Let us prove first part

(X+Y).X'.Y' = 1

(X +Y) + X'Y' = ((X+Y) + X').((X+Y)+Y')    [(X+Y)(X+Z) = X+Y.Z]

=(X+X'+Y).(X+Y+Y')

=(1+Y)(X+1)         [X+X' = 1inverse law] 

= 1.1 = 1             [1 +X = 1identity law ]