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प्रश्न
State DeMorgan's Laws of Boolean Algebra and verify them using a truth table.
उत्तर
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 ]
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DeMorgan’S Law/Theorem and Their Applications
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