laws (theorems) of boolean algebra laws of complementation othe term complement means, to invert or...

Laws (Theorems) of Boolean algebra

Laws of Complementationo 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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QUESTION PAPERS 1

Law 1: = 1

Law 2: = 0

Law 3: If A = 0,

Law 4: If A = 1,

Law 5:

0

1

1A

0A

AA

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QUESTION PAPERS 2

AND Laws

• Law 6: A.0=0

• Law 7: A.1=A

• Law 8: A.A=A

• Law 9: 0. AA

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QUESTION PAPERS 3

OR Laws

• Law 10: A +0 = A

• Law 11: A +1 = 1

• Law 12: A +A = A

• Law 13: 1 AA

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QUESTION PAPERS 4

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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QUESTION PAPERS 5

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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QUESTION PAPERS 6

• Law 17: A.(B.C) = (A.B).C

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QUESTION PAPERS 7

Distributive LawsThese 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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QUESTION PAPERS 8

De Morgan's Theorems

• 1.

• 2.

• The complement of any Boolean expression is found by using these two rules.

YXYX

YXYX

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QUESTION PAPERS 9

Steps for complementation:

• 1. Replace ‘+’ symbols with ‘.’ symbols and ‘.’ symbols with ‘+’ symbols.

• 2. Complement each term.

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QUESTION PAPERS 10

Proof of De Morgan's Theorems

YXYX

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QUESTION PAPERS 11

YXYX

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QUESTION PAPERS 12