The Algebra of Boolean

One of the primary requirements when dealing with digital circuits is to find ways to make them as simple as possible. This constantly requires that complex logical expressions be reduced to simpler expressions that nevertheless produce the same results under all possible conditions. The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates.
One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra. The rules of Boolean Algebra are simple and straight-forward, and can be applied to any logical expression. The resulting reduced expression can then be readily tested with a Truth Table, to verify that the reduction was valid.
The rules of Boolean Algebra are:.

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AND Operations (·)

0·0 = 0      A·0  = 0
1·0 = 0      A·1  = A
0·1 = 0      A·A  = A
1·1 = 1      A·A' = 0

OR Operations (+)

0+0 = 0      A+0  = A
1+0 = 1      A+1  = 1
0+1 = 1      A+A  = A
1+1 = 1      A+A' = 1

NOT Operations (')

0' = 1       A''  = A
1' = 0

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Associative Law

(A·B)·C = A·(B·C) = A·B·C
(A+B)+C = A+(B+C) = A+B+C

Distributive Law

A·(B+C) = (A·B) + (A·C)
A+(B·C) = (A+B) · (A+C)

Commutative Law

A·B = B·A
A+B = B+A

Precedence

AB = A·B
A·B+C = (A·B) + C
A+B·C = A + (B·C)

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DeMorgan's Theorem

(A·B)' = A' + B'   (NAND)

(A+B)' = A' · B'   (NOR)