Boolean Algebra

Boolean algebra is the set of rules or laws that help in simplifying any logical expression. In Boolean algebra, there can be a large number of variable in a logic expression but the only value each variable holds is either 1 or 0.  When a number of variables is less (i.e. 2 or 3) then Boolean algebra is effective.

Laws Of Boolean Algebra

• Complement Law (Inversion Law) 

A’ = Complement of A

(A’)’ = A

• AND Law 

A.0 = 0

A.1 = A

A.A = A

A.A’ = A

• OR Law 

A + 0 = A

A + 1 = 1

A + A = A

A + A’ = 1

• Commutative Law 

A + B = B + A

A.B = B.A

• Associative Law 

(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 + B.C

• DeMorgan’s Law 

(A + B)’ = A’.B’

(A.B)’ = A’ + B’

• Other Important Laws

A + AB = A

A’ + AB = A’ + B

A + A’B = A + B

A’ + AB’ = A’ + B’

How to simplify a boolean expression

Q1. Simplify the following Boolean expression:

A.B.C + A’ + A.B’.C

Here we have to simplify the above boolean expression, hence we will use the laws of boolean algebra to simplify it.

A.B.C + A’ + A.B’.C

A.C.(B + B’)  +  A’                           [By distributive law]

A.C.(1)  + A’                                     [By OR law]

A.C + A’                                           [By AND law]

A’ + C                                               [A’ + AB = A’ + B' ]

Q2. Find the boolean expression for the given logic circuit:

Here we have been given a logic circuit and we have to find the final boolean expression for it.

So, at first we consider the OR Gate. In OR Gate we have three inputs, A, B, C. Hence its output is A+B+C. Now, we consider the NOT Gate. D is the input of NOT Gate. Hence, its output is D'.

The last of the circuit is the AND Gate in which there goes two inputs (A+B+C) and D'. So the final output is (A+B+C).D' .