2.3 Simplication of Boolean Equation

There are two ways to simplify Boolean equation, Laws of Boolean Algebra and Karnaugh Map:

Laws of Boolean Algebra

Operations on Constants
X.1 = X
X.0 = 0
X+1 = 1
X+0 = 0

Idempotent Law
X.X = X
X+X=X

Absorption Law
X.(Y.Z) = (X.Y).Z

Double Complement Law

Laws of Commutativity
X.Y = Y.X
X+Y = Y+X

Laws of Associativity
X.(Y.Z) = (X.Y).Z
X+(Y+Z) = (X+Y)+Z

Laws of Distributivity
X.(Y+Z) = (X.Y)+(X.Z)
X+(Y.Z) = (X+Y).(X.Z)

Inverse laws
X.~X = 0
X+~X = 1
~~X = X

DeMorgan’s laws
~(X.Y) = ~X+~Y
~(X+Y) = ~X.~Y


Karnaugh Map

Karnaugh map (K-map) is a convenient way to represent Boolean function, the only limitation is that it will be ineffective for > 4 inputs. It is essential for later purposes to list the combinations in the order 00, 01, 11 10. Because the squares corresponding to the combinations are to be used for recording information, the combinations are customarily written above the squares.

K-map can be used to represent any Boolean function in the following way.

Each square corresponds to a unique product in the sum-of-products (SOP) form, with a 1 value corresponding to the variable and a 0 value corresponding to the NOT of that variable. For each such product in the function, 1 is placed in the corresponding square.

Once the map of a function is created, we can often write a simple algebraic expression for it by noting the arrangement of the 1s on the map.The principle is as follows. Any two squares that are adjacent differ in only one of the variables. If two adjacent squares both have an entry of one, then the corresponding product terms differ in only one variable. In such a case, the two terms can be merged by eliminating that variable. For example, A'BC'D + A'BCD = A'BD. This process can be extended in several ways.