Laws (Theorems) of Boolean algebra Laws of Complementation oThe term complement means, to invert or to change 1's to 0's and 0's to 1's, for which purpose.

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Laws (Theorems) of Boolean algebra Laws of Complementation oThe 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. oA complement of a variable is represented by a bar over the letter. For example, the complement of a variable A will be denoted by | Website for Students | VTU NOTES | QUESTION PAPERS1

Law 1: = 1 Law 2: = 0 Law 3: If A = 0, Law 4: If A = 1, Law 5: | Website for Students | VTU NOTES | QUESTION PAPERS2

AND Laws Law 6: A.0=0 Law 7: A.1=A Law 8: A.A=A Law 9: | Website for Students | VTU NOTES | QUESTION PAPERS3

OR Laws Law 10: A +0 = A Law 11: A +1 = 1 Law 12: A +A = A Law 13: | Website for Students | VTU NOTES | QUESTION PAPERS4

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 | Website for Students | VTU NOTES | QUESTION PAPERS5

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 | Website for Students | VTU NOTES | QUESTION PAPERS6

Law 17: A.(B.C) = (A.B).C | Website for Students | VTU NOTES | QUESTION PAPERS7

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) | Website for Students | VTU NOTES | QUESTION PAPERS8

De Morgan's Theorems The complement of any Boolean expression is found by using these two rules. | Website for Students | VTU NOTES | QUESTION PAPERS9

Steps for complementation: 1. Replace ‘+’ symbols with ‘.’ symbols and ‘.’ symbols with ‘+’ symbols. 2. Complement each term. | Website for Students | VTU NOTES | QUESTION PAPERS10

Proof of De Morgan's Theorems | Website for Students | VTU NOTES | QUESTION PAPERS11

| Website for Students | VTU NOTES | QUESTION PAPERS12