Question

Give a circuit that computes the parity function on three input variables and show how it computes on input 011.

Short answer

Using the Parity function definition and the truth tables of AND & OR operators we can solve the above problem.

Step-by-step solution

Understanding the definition of the Parity Function

The Boolean function $f\text{:}{\{0,1\}}^n\to \{0,1\}$gives an output 1 if an odd number of 1’s appear in the input variable otherwise gives an output 0. Where we can define f as:$f\left(x\right)=x_1\oplus x_2\oplus x_3\oplus \dots \dots \oplus x_n$

, where$\oplus$ is denoted as an EX-OR Gate operation.

Since a number of gates are used in this parity functional circuit, the working of gates will first be discussed.

Understanding the working of AND operator 

The Truth table of AND gate

Circuit Diagram of AND operator

Understanding of the NOT gate

The Truth table of NOT gate

INPUT(x)	OUTPUT(y)
0	1
1	0

Circuit Diagram of NOT operator

Understanding of OR gate 

The truth table:

X	Y	OUTPUT
0	0	0
1	0	1
0	1	1
1	1	1

Circuit Diagram of OR operator

Defining the parity function for input variables

For three input variables, the parity function is defined as:

$\begin{array}{c}f\left(x\right)=x_1\oplus x_2\oplus x_3\\ =[(x_1.{\overline{x}}_2)+({\overline{x}}_1.x_2)]\oplus x_3\\ -[x_1.{\overline{x}}_2.{\overline{x}}_3+{\overline{x}}_1.x_2.{\overline{x}}_3+x_1.x_2.{\overline{x}}_3+{\overline{x}}_1.{\overline{x}}_2.x_3]\end{array}$

Take into consideration the below diagram:

Computing the three-input parity function

Perform computation of the three-input parity functional circuit by taking 001 as an input and apply an EX-OR operation (EX-OR gate consists of an AND, OR and NOT hate operations) using the parity function for three input variables definition.

The output will be 0, because of the even number of 1’s in the input. (By the definition of parity functional circuit)

The below diagram shows the, computation of 3 input parity functional circuit, taking 001 as input.