Boolean Function

 

Part 1: What is Boolean Function?

 

Boolean algebra is a system of mathematics that works with true/false values (1s and 0s) and logical operations like AND, OR, and NOT. A Boolean function is a mathematical expression that uses these values and operations to produce a true (1) or false (0) result based on the input values. Boolean functions can be defined in three primary ways: Boolean Expression, Truth Table, and Logic Circuit.

 

 

Part 2. Boolean Expression.

 

Boolean expressions are mathematical expressions that use Boolean variables (A, B, C, etc.) and Boolean operators (AND, OR, NOT, XOR, etc.) to represent logical relationships. They are the foundation of digital logic and are used to describe the behavior of digital circuits.

 

For example, the Boolean Expression:

 

F = x + y’z

 

 

Part 3. Truth Table

 

A truth table is a tabular representation of a Boolean function. It provides a clear and concise way to visualize the relationship between the input values and the corresponding output value of a Boolean expression. Each row in a truth table represents a unique combination of input values, while each column represents an input variable or the output of the function. The values in the table are typically 0 or 1, representing the logical values "false" and "true," respectively. Truth tables are essential for understanding Boolean functions, verifying the correctness of Boolean expressions, and designing digital circuits.

 

Part 3.1 How we Draw a Truth Table

 

This is the step by step procedure how we draw a truth table.

 

Step No. 1: Know the number of input. The use the formula below to know the number of combination you need to do.

Number of combination = 2n

 

Step No. 2 Draw the row of input, output and combination.

 

Step No. 3 Draw the combinations. Use the pattern.

 

Example no 1.

Draw a truth table with 3 input and 1 output variable.

 

Solution:

Step No. 1: Know the number of input. The use the formula below to know the number of combination you need to do.

Number of combination = 2n

Number of combination = 23

Number of combination = 8

 

Step No. 2 Draw the row of input, output and combination.

 

Input

Output

X

Y

Z

F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step No. 3 Draw the combinations. Use the pattern.

 

Input

Output

X

Y

Z

F

0

0

0

 

0

0

1

 

0

1

0

 

0

1

1

 

1

0

0

 

1

0

1

 

1

1

0

 

1

1

1

 

 

And here is our truth table.

 

Example no 2.

Draw a truth table with 4 input and 1 output variable.

 

Solution:

Step No. 1: Know the number of input. The use the formula below to know the number of combination you need to do.

Number of combination = 2n

Number of combination = 24

Number of combination = 16

 

Step No. 2 Draw the row of input, output and combination.

 

Input

Output

w

x

y

z

F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step No. 3 Draw the combinations. Use the pattern.

 

Input

Output

w

x

y

z

F

0

0

0

0

 

0

0

0

1

 

0

0

1

0

 

0

0

1

1

 

0

1

0

0

 

0

1

0

1

 

0

1

1

0

 

0

1

1

1

 

1

0

0

0

 

1

0

0

1

 

1

0

1

0

 

1

0

1

1

 

1

1

0

0

 

1

1

0

1

 

1

1

1

0

 

1

1

1

1

 

 

And here is our truth table.

 

Part 4. Logic Circuit

 

Logic circuits are electronic circuits that implement Boolean functions. They are the building blocks of digital systems and are composed of logic gates, such as AND, OR, NOT, NAND, NOR, and XOR. These gates perform specific logical operations on binary inputs (0s and 1s) to produce a binary output. By combining logic gates in various ways, complex Boolean functions can be implemented. Logic circuits are used in a wide range of applications, including digital computers, microprocessors, communication systems, and control systems.

 

Example of Logic Circuit.

 

 

 

Part 5. Conversion from Boolean Expression to Truth Table.

 

This is the step by step procedure how we convert Boolean expression to truth table.

Step No. 1: Know the number of input variables. The use the formula below to know the number of combination you need to do.

Number of combination = 2n

Step No. 2 Draw the row of input, output and combination.

Step No. 3 Draw the combinations. Use the pattern.

Step No. 4 Evaluate each term of the expression.

 

Example No 3.

 

Convert F=ABC+BC’+AC’ to truth table

 

Solution:

Step No. 1: Know the number of input variables. The use the formula below to know the number of combination you need to do.

Number of combination = 2n

Number of combination = 23

Number of combination = 8

 

Step No. 2 Draw the row of input, output and combination.

 

Input

Output

A

B

C

F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step No. 3 Draw the combinations. Use the pattern.

 

Input

Output

A

B

C

F

0

0

0

 

0

0

1

 

0

1

0

 

0

1

1

 

1

0

0

 

1

0

1

 

1

1

0

 

1

1

1

 

 

Step No. 4 Evaluate each term of the expression.

 

Step No. 4.1

F=ABC+BC’+AC’

Input

Output

A

B

C

C’

0

0

0

1

0

0

1

0

0

1

0

1

0

1

1

0

1

0

0

1

1

0

1

0

1

1

0

1

1

1

1

0

 

Step No. 4.2

F=ABC+BC’+AC’

Input

Output

A

B

C

C’

ABC

0

0

0

1

0

0

0

1

0

0

0

1

0

1

0

0

1

1

0

0

1

0

0

1

0

1

0

1

0

0

1

1

0

1

0

1

1

1

0

1

 

Step No. 4.3

F=ABC+BC’+AC’

Input

Output

A

B

C

C’

ABC

BC’

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

1

0

1

0

1

1

0

0

0

1

0

0

1

0

0

1

0

1

0

0

0

1

1

0

1

0

1

1

1

1

0

1

0

 

Step No. 4.4

F=ABC+BC’+AC’

Input

Output

A

B

C

C’

ABC

BC’

AC’

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

1

0

1

0

0

1

1

0

0

0

0

1

0

0

1

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

1

1

1

1

1

0

1

0

0

 

 

Step No. 4.4

F=ABC+BC’+AC’

Input

Output

A

B

C

C’

ABC

BC’

AC’

ABC+BC’+AC’

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

1

0

1

0

1

0

1

1

0

0

0

0

0

1

0

0

1

0

0

1

1

1

0

1

0

0

0

0

0

1

1

0

1

0

1

1

1

1

1

1

0

1

0

0

1

 

 

Step No. 4.5

Our Final truth table is:

F=ABC+BC’+AC’

Input

Output

A

B

C

F

0

0

0

0

0

0

1

0

0

1

0

1

0

1

1

0

1

0

0

1

1

0

1

0

1

1

0

1

1

1

1

1

 

 

Part 6. Conversion from Boolean Expression to Logic Circuit.

 

To convert a Boolean expression to a logic circuit, follow these steps:

 

Step 1. Identify the variables: Determine the distinct variables present in the expression.

Step 2. Create a logic gate for each operation: Use logic gates (AND, OR, NOT, NAND, NOR, XOR, XNOR) to represent each Boolean operation in the expression.

Step 3. Connect the gates: Connect the input and output pins of the gates according to the order of operations in the expression.

Step 4. Label the inputs and outputs: Label the input pins with the corresponding variables and the output pin with the name of the expression.

 

Example no 4.

 

Convert F=ABC+BC’+AC’ to logic circuit.

Solution:

Step 1. Identify the variables: Determine the distinct variables present in the expression.

Inputs: A, B, and C

Output: F

 

Step 2. Create a logic gate for each operation: Use logic gates (AND, OR, NOT, NAND, NOR, XOR, XNOR) to represent each Boolean operation in the expression.

 

 

Step 3. Connect the gates: Connect the input and output pins of the gates according to the order of operations in the expression.

 

Step 4. Label the inputs and outputs: Label the input pins with the corresponding variables and the output pin with the name of the expression.

 

 

 

Part 6. Conversion from Logic Circuit to Boolean Expression.

 

To convert a logic circuit to a Boolean expression, follow these steps:

 

Step 1. Identify the logic gates: Determine the types of logic gates (AND, OR, NOT, NAND, NOR, XOR, XNOR) used in the circuit.

Step 2. Label the inputs and outputs: Label the input and output pins of each gate with variables or expressions.

Step 3. Write Boolean expressions for each gate: Write the Boolean expression corresponding to each logic gate, using the labeled inputs and outputs.

Step 4. Combine expressions: Combine the expressions for the individual gates to form the overall Boolean expression for the circuit.

 

Example no 5.

 

Convert the Logic Circuit below to Boolean Expression.

 

 

Solution:

 

 

Our answer is F = ABC + AC’ + BC’.