The Laws of Boolean Algebra
Boolean expressions can be simplified using a set of algebraic laws in much the same way as ordinary algebraic expressions.
A set of rules formulated by the English mathematician George Boole describe certain propositions whose outcome would be either true or false. With regard to digital logic, these rules are used to describe circuits whose state can be either, 1 (true) or 0 (false). In order to fully understand this, the operations of the AND, OR and NOT gates are stated.
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0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 1 | 0 |
The Boolean Laws
Note that every law has two expressions, (a) and (b). This is known as duality. These are obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to 0's and vice-versa.It has become conventional to drop the . (AND symbol) i.e. A.B is written as AB.
T1 : Commutative Law
(a)
(b)
T2 : Associate Law
(a)
(b)
T3 : Distributive Law
(a)
(b)
T4 : Identity Law
(a)
(b)
T5 :
(a)
(b) 
T6 : Redundance Law
(a)
(b)
T7 :
(a)
(b)
T8 :
(a)
(b)
T9 :
(a)
(b) 
T10 :
(a)
(b) 
T11 : De Morgan's Theorem
(a)
(b) 
