Transcript Laws (Theorems) of Boolean algebra

Laws (Theorems) of Boolean algebra
Laws of Complementation
o The term complement means, to invert or to
change 1's to 0's and 0's to 1's, for which
purpose inverters or NOT gates are used.
o A complement of a variable is represented by a
bar over the letter. For example, the
complement of a variable A will be denoted by
A
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Law 1: 0 = 1
Law 2: 1 = 0
Law 3: If A = 0, A  1
Law 4: If A = 1, A  0
Law 5: A  A
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AND Laws
• Law 6: A.0=0
• Law 7: A.1=A
• Law 8: A.A=A
• Law 9: A.A  0
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OR Laws
• Law 10: A +0 = A
• Law 11: A +1 = 1
• Law 12: A +A = A
• Law 13:
A A 1
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Commutative Laws
– This states that the order in which the variables
are OR’ed and AND’ed will make no
difference in the output.
• Law14: A. B = B. A
• Law 15 : A + B = B + A
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Associative Laws
 This law states that the order in which the
variables are grouped will not make any difference
in the output.
•
Law 16: A + (B + C) = (A + B) + C
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• Law 17: A.(B.C) = (A.B).C
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Distributive Laws
These laws allow the factoring or multiplying out of
expressions.
• Law 18: A .(B +C) = (A .B) + (A .C)
• Law 19: A + (B .C) = (A + B) (A + C)
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De Morgan's Theorems
• 1.
X  Y  X Y
• 2.
X Y  X  Y
• The complement of any Boolean expression is
found by using these two rules.
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Steps for complementation:
• 1. Replace ‘+’ symbols with ‘.’ symbols and ‘.’
symbols
with ‘+’ symbols.
• 2. Complement each term.
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Proof of De Morgan's Theorems
X  Y  X Y
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X Y  X  Y
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