Boolean Algebra (Operation, Laws, Calculation, Truth Table)

Boolean algebra is a widely used concept in mathematics, electronics, and machine language to tell true and false values such as 0 & 1. All the digital circuits and digital gates are analyzed by this category of algebra.

There are vast applications of Boolean algebra in our daily life. In this article, we’ll learn the basics of Boolean algebra such as what is it, its theorems, gates, truth table, and solved examples. 

What is Boolean Algebra?

Boolean algebra is a well-known branch of algebra that is used to simplify and analyze digital circuits or digital gates. This branch of algebra is also known as binary algebra or logical algebra. The binary and logical words are used because Boolean algebra involved binary variables 0 & 1 and deals with the logical gates. 

All the electronics development is based on Boolean algebra as the development is based on turning on and turning off the electronics and for this binary variables and logical gates are used that are the essential factors of Boolean algebra. 

For the visual representation of the binary algebra, the Venn diagram is a key factor. The binary variables used in Boolean algebra such as 0 and 1 denote false and true values respectively. And logical operations are used to perform addition and subtraction. 

What are the operations of Boolean algebra?

There are three well-known operations of binary algebra that are used to perform various operations on the binary values. 

These three operations are:

1. Conjunction

2. Disjunction

3. Negation 

Conjunction, disjunction, and negation are used to perform arithmetic and numeric operations on binary variables like addition and subtraction. These operations are known as Boolean operators.

Boolean Operators	Notation	Representation	Definition
Conjunction	^	AND	In this operation, the values are true when both the terms are true otherwise false. It acts as a product of numbers. A.B or A^B
Disjunction	v	OR	In this operation, the values are false when both the terms are false or otherwise true. It acts as a product of numbers. A+B or AvB
Negation	¬	NOT	It reverts the binary variables such as if true it transposes it to false and if false it converts it to true.

Read Also: Difference Between Combinational and Sequential Circuit

What is a truth table in Boolean Algebra?

Truth tables are mathematical tables used to determine whether a compound statement is true or false. There are typically two columns in a truth table each listing all the possible truth values for each statement. Each statement is assigned a letter or variable, such as a, b, or c.

A truth table for conjunction

A	B	A.B or A^B
1	1	1
1	0	0
0	1	0
0	0	0

A truth table for disjunction

A	B	A+B or AvB
1	1	1
1	0	1
0	1	1
0	0	0

A truth table for negation

A	¬A
1	0
0	1

Read Also: What are Decoders? Block Diagram, Truth Table, Types

Theorems of Boolean algebra

There are two well-known theorems of Boolean algebra such as:

1.De Morgan’s 1st law

2.De Morgan’s 2nd law

In order to change the Boolean expression, these two theorems must be used. The basic idea behind this theorem is to simplify the given Boolean expression. It is possible to change the form of an expression by using these two De Morgan laws.

Theorems	Statement	Expression
De Morgan’s 1st law	The 1st law of De Morgan states that the complement of the product of the variables is equal to the sum of their individual complements of a variable.	(X.Y)’ = X’+Y’
De Morgan’s 2nd law	The 2nd law of De Morgan states that the complement of the sum of variables is equal to the product of their individual complements of a variable.	(X+Y)’ = X’.Y’

The truth table of De Morgan’s 1st Law

X	Y	X’	Y’	X.Y	(X.Y)’	X’ + Y’
0	0	1	1	0	1	1
0	1	1	0	0	1	1
1	0	0	1	0	1	1
1	1	0	0	1	0	0

Hence from the last two columns De Morgan’s 1st law holds.

(X.Y)’ = X’+Y’

The truth table of De Morgan’s 2nd Law

X	Y	X’	Y’	X + Y	(X + Y)’	X’ * Y’
0	0	1	1	0	1	1
0	1	1	0	1	0	0
1	0	0	1	1	0	0
1	1	0	0	1	0	0

Hence from the last two columns De Morgan’s 1st law holds.

(X + Y)’ = X’.Y’

Read Also: Comparison between RTL, DTL, and TTL Logic Families

Laws of Boolean Algebra

Here are some well-known laws of Boolean algebra. 

Laws Name	Definition	Expression
Commutative Law	As a result of commutative law, a logic circuit's output does not change if its variables are changed in sequence.	X * Y = Y * X X + Y = Y + X
Associative Law	According to this law, the order in which logic operations are performed has no effect on their effects.	(X * Y) * Z = X *(Y*Z) (X + Y) + Z = X+(Y+Z)
Distributive Law	This law is used for both addition and multiplication and states that	X* (Y + Z) = (X* Y) + (X* Z) X+ (Y * Z) = (X+ Y) * (X+ Z)
AND Law	The law that uses AND operation is said to be the AND law of binary algebra.	X * 0 = 0 X * 1 = X X * X = X X * x̄=0
OR Law	The law that uses the OR operation is said to be the OR law of binary algebra.	X + 0 = X X + 1 = 1 X + X = X X +x̄= 1
Inversion Law	The inversion law of Boolean algebra states that double inversion of the original variable produces the original variable.	x̄̄ = x

Read Also: [Actual] Difference Between Latch and Flip Flop Explained

How to calculate Boolean algebra problems?

The Boolean algebra problems can be solved in two ways such as

1. By using laws of Boolean algebra

2. By using the truth table 

Let us take a few examples to solve Boolean algebra problems according to both methods. 

Example 1

Evaluate the given expression of the binary algebra with the help of laws of Boolean algebra and truth table. 

(X +Y) + (X * Z) *(X+Z)

Solution 

Step 1: First of all, take the given binary algebra expression.

(X + Y) + (X * Z) * (X + Z)

Now evaluate the above binary algebra expression with the help of the laws of Boolean algebra. 

Step 2: Use the sum of the product law of the Boolean algebra.

(X + Y) + (X * Z) * (X + Z)= (X + Y) + XZ * (X + Z)

(X + Y) + (X * Z) * (X + Z) = X + Y + XXZ + XZZ

Step 3: Now use the Idempotent Law of Boolean algebra (XX = X) to the above expression.

(X + Y) + (X * Z) * (X + Z) = X + Y + XZ + XZ

Step 4: Factorize the above expression.

(X + Y) + (X * Z) * (X + Z) = X + Y +X(Z + Z)

Step 5: Now use the idempotent law with respect to the sum of the above expression (X + X = X). 

(X + Y) + (X * Z) * (X + Z) = X + Y + XZ

(X + Y) + (X * Z) * (X + Z) = Y + (Z + 1)X

Step 6: Now use the identity law of binary algebra to the above expression. 

(X + Y) + (X * Z) * (X + Z) = Y + (1)X

(X + Y) + (X * Z) * (X + Z) = Y + X

(X + Y) + (X * Z) * (X + Z) = X + Y

Alternatively

Now solve the given binary expression with the help of the truth table.

Step 1: First of all, take the number of variables to the power of 2 for getting the number of rows of the truth table. 

Terms in the given expression = n = 3

According to formula 

2n = 23 = 2 x 2 x 2 = 8

Hence, there will be 8 rows in the truth table.

Step 2: Now make a truth table according to the given expression and get the result.

X	Y	Z	X + Y	X * Z	X + Z	(X + Y) + (X * Z)	(X + Y) + (X * Z) * (X + Z)
0	0	0	0	0	0	0	0
0	0	1	0	0	1	0	0
0	1	0	1	0	0	1	0
0	1	1	1	0	1	1	1
1	0	0	1	0	1	1	1
1	0	1	1	1	1	1	1
1	1	0	1	0	1	1	1
1	1	1	1	1	1	1	1

The Boolean algebra calculator by AllMath can be used to solve the problems of binary algebra either with the help of law or a truth table with steps. 

Example 2

Evaluate the given expression of the binary algebra with the help of laws of Boolean algebra and truth table.

[(X+Y) + Z]+[X * (Y+Z)]

Solution 

Step 1: First of all, take the given binary algebra expression.

[(X + Y) + Z] + [X * (Y + Z)]

Now evaluate the above binary algebra expression with the help of the laws of Boolean algebra. 

Step 2: Use the sum of the product law of the Boolean algebra.

[(X + Y) + Z] + [X * (Y + Z)] = [X + Y + Z] + [XY + XZ]

[(X + Y) + Z] + [X * (Y + Z)] = X + Y + Z + XY + XZ

Step 3: Factorize the above expression.

[(X + Y) + Z] + [X * (Y + Z)] = (1 + Y)X + Y + Z + XZ

[(X + Y) + Z] + [X * (Y + Z)] = (1 + Y)X + Y + (1 +X) Z

Step 5: Now use the identity law with respect to the sum of the above expression (1 + X = 1)& (1 + y = 1). 

[(X + Y) + Z] + [X * (Y + Z)] = (1) X + Y + (1) Z

[(X + Y) + Z] + [X * (Y + Z)] = X + Y + Z

Alternatively

Now solve the given binary expression with the help of the truth table.

Step 1: First of all, take the number of variables to the power of 2 for getting the number of rows of the truth table. 

Terms in the given expression = n = 3

According to formula 

2n = 23 = 2 x 2 x 2 = 8

Hence, there will be 8 rows in the truth table.

Step 2: Now make a truth table according to the given expression and get the result.

X	Y	Z	X + Y	Y + Z	(X + Y) + Z	X * (Y + Z)	[(X + Y) + Z] + [X * (Y + Z)]
0	0	0	0	0	0	0	0
0	0	1	0	1	1	0	1
0	1	0	1	1	1	0	1
0	1	1	1	1	1	0	1
1	0	0	1	0	1	0	1
1	0	1	1	1	1	1	1
1	1	0	1	1	1	1	1
1	1	1	1	1	1	1	1

Sum Up

Boolean algebra is a technique used in digital circuits and electronic circuits for various purposes. Now you can grab all the basics of Boolean algebra from this article as we have discussed all the basics of this post with solved examples. You can learn the definition, operations, laws, theorems, and solved examples from this post. 

Read Also: 

• [BEST] Application and Uses of Logic Gates with Practical Examples
• Truth Table and applications of all types of Flip Flops-SR, JK, D, T, Master-Slave

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