Intro to COA

# Boolean Expressions And Functions

Boolean algebra developed by George Boole in 1847. Before starting the topic, look at the basic terms (i.e., Boolean Algebra, Boolean Expressions, and Boolean Functions).

• A Boolean Algebra deals with binary variables (i.e., A, B, C), Binary Constants (0,1), and logic operations (i.e., NOT, AND, OR).
• Boolean function is described by an algebraic expression called Boolean expression.
• Boolean expression consists of binary variables, the constants (0 and 1), and the logic operation symbols.

The Boolean Expression is an INPUT, and the Boolean Function is an OUTPUT.

Note: The Output of the Boolean algebra function on the given value of variables can be TRUE (1) or FALSE(0). Let’s explain the following Boolean functions.

## Symbolic Representation Of Boolean Expressions/Functions

Symbolically, the Boolean functions can be represented through Logic gates. Consider the function  F = (x.y) + (y’.z). The symbolic representation of this function is given below.

## OUTPUT of Boolean Expressions/Functions

To find the Output of any Boolean function, the Binary value of each variable is already given as an INPUT. SO replace the variables with their respective input values.

Let’s suppose a Boolean function F(x,y,z) = (x.y) + (y’.z) and its input values are X=0, Y=1, Z=0, then the values of the Boolean function. The Output of the given function is

``` F(x,y,z) = (x.y) + (y'.z)
As Y= 1 then, Y’ = 0
F(x,y,z)= (0.1) + (0.0)
F(x,y,z)= (0) + (0)
F(x,y,z)= 0```

## Truth Table of Boolean Function

To represent a function in a truth table, we require a list of the 2(N) combinations, where N is the number of binary variables. If there are three variables (x, y, and z), then 2(3) = 8 combinations will required. For a better understanding of the Boolean Truth table, look at the topic of logic gates.

The truth table for the Boolean function F = (x.y) + (y’.z) is given under

## Boolean Functions with Truth Tables

Let’s explain some examples of Boolean Algebra Functions with their Truth Tables.

### Example 01: F(X,Y)= (X+Y). Z

Block diagram for the Boolean function F(X, Y)= (X+Y). Z

The truth table for the Boolean function F(X.Y)= (X+Y).Z

### Example 02: F(X,Y)= (X’.Y’) + (X’.Z)

Block diagram for the Boolean function F(X,Y)= (X’.Y’) + (X’.Z)

The truth table for the Boolean function F(X,Y)= (X’.Y’) + (X’.Z)

### Example 03: F(X,Y) = (x’y’z) + (x’yz) + (xy’)

Block diagram for the Boolean function F(X,Y) = (x’y’z) + (x’yz) + (xy’)

The truth table for the Boolean function F(X,Y) = (x’y’z) + (x’yz) + (xy’)