Boolean Expression/Function


Boolean algebra deals with binary variables and logic operation. A Boolean Function is described by an algebraic expression called Boolean expression which consists of binary variables, the constants 0 and 1 and the logic operation symbols. Consider the following example

Boolean Function

Here the left side of the equation represents the output Y. So we can state equation no. 1

Boolean Function1

Truth Table Formation

A truth table represents a table having all combinations of inputs and their corresponding result.

It is possible to convert the switching equation into a truth table. For example consider the following switching equation.

Switching Equation

The output will be high (1) if A = 1 or BC = 1 or both are 1. The truth table for this equation is shown by Table (a). The number of rows in the truth table is 2n where n is the number of input variables (n=3 for the given equation). Hence there are 23 = 8 possible input combination of inputs.

Truth Table

Methods to simplify the boolean function

The methods used for simplifying the Boolean function are as follows.

  • Karnaugh-map or K-map.

  • NAND gate method

Karnaugh-map or K-map

The Boolean theorems and De-Morgan's theorems are useful in manipulating the logic expression. We can realize the logical expression using gates. The no. of logic gates required for the realization of a logical expression should be reduced to minimum possible value by K-map method.This method can be by two way

Sum of Products (SOP) Form

It is in the form of sum of three terms AB,AC,BC with each individual term is product of two variable. Say A.B or A.C etc. Therefore such expression are known as expression in SOP form. The sum and products in SOP form are not the actual additions or multiplications. In fact they are the OR and AND functions. In SOP form, 0 is represent for bar and 1 is represent for unbar. SOP form is represented by SOP.

Example of SOP is as follows.

SOP example

Product of Sums (POS) Form

It is in the form of product of three terms (A+B),(B+C) and (A+C) with each term is in the form of sum of two variables. Such expression are said to be in the product of sums (POS) form. In POS form, 0 is represent for unbar and 1 is represent for bar. POS form is represented by POS.

Example of POS is as follows.

POS example

NAND gates Realization

NAND gates can be used to simplify boolean functions as shown in example below.

NAND example