**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’)