Boolean Algebra

Boolean algebra laws

There are following laws in Boolean algebra:

Associative Law: This law states that if we have 3 variables x, y, z then

X*(Y*Z) = (X*Y)*Z

Commutative Law: This law states that

X*Y = Y*X

Identity element: If e is the identity then we have the relation with the Boolean algebra

e*x = x*e=x Hence 0 is identity for + as x + 0 = x = 0 + x & 1 is for dot(.) as x.1=x=1.x

Compliment: x+x’=1      x.x’=0

Boundedness Law: x+1= 1           x.0=0

Distributive law: Suppose we have two binary operators * and . then this law states that

X+(y.z) = (x+y) .  (x+z)                    x.(y+z)= x.y + x.z

DeMorgan Law:                (x + y)’ = x’. y’                                    (x.y)’ = x’ + y’

This can also be generalized and stated as that whenever we want to take compliment of any function we just have to take compliment of each literal and change AND to OR & OR to AND and 0 to 1 & 1 to 0

(A+B+C+D+E…. +Z)’ = A’ B’ C’ D’ E’ …..Z’

(ABCDE….Z)’ = A’ + B’ + C’ +D’ + E’……+Z’

Absorption law:               x + xy = x                                             x(x+y)=x

Elimination law:               x + (x’.y) = x + y                                                x.(x’ + y)=x.y

Unique compliment theorem:  If we have x + y =1 and x . y = 0, then x = y’

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