What is Boolean Algebra? Its definition, explanation, and examples

What is Boolean algebra Its definition
What is Boolean algebra Its definition

Boolean algebra is a frequently used method in the language of the computer to encode various kinds of terms. This method is usually used in electronics and computer languages. It is a sub-branch of algebra that is used either in mathematics or computer.

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This method used binary variables such as 0 and 1 for denoting and expressing various terms. In this article, we are going to explain Boolean algebra along with its definition, explanation, and examples. 

Boolean algebra (Binary algebra)

The division of mathematics that deals with different operations on logical values and includes binary variables 0 & 1 is known as binary algebra (Boolean algebra). The binary variables play a vital role in binary algebra to encode the symbols in machine language.  

The binary variables are used to express whether the statement is true or false. The binary number 1 is used for the true statement and 0 for the false statement. In electronics, when the switch is on it is true and when the switch is off then it is false.   

Operations of Binary Algebra

Here are the following operations of Boolean algebra. 

  • Conjunction 
  • Disjunction
  • Negation 
  • Nand 
  • XOR 

Arithmetic operations such as multiplication and addition are used in binary algebra with the help of Boolean algebra gates (operations). The working of addition and multiplication is different with respect to actual sum and product. 

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The truth tables are used to describe the operation of binary algebra. Let us discuss the operations of binary algebra along with truth tables. 

  • Conjunction 

The conjunction operation of binary algebra is the multiplication operation. The product of two binary variables is 1 when both the digits are 1 otherwise it gives 0. In other words, when both the terms are true then the product is true otherwise it is false. 

AND operation is another name for conjunction as multiplication is involved in this gate. The “^” cap symbol denotes the conjunction operation of binary algebra and is written as P ^ Q. Let us have two digits such as P & Q then the product operation is applied with the help of the truth table.

The number of rows is decided by the total number given in the expression, as P & Q are two numbers then the number of rows is 22 = 4.

PQP ^ Q
1
1
111
  • Disjunction

The disjunction operation of binary algebra is the addition operation. The sum of two binary variables is 0 when both the digits are 0 otherwise it gives 1. In other words, when both the terms are false then the sum is false otherwise it is true. 

OR operation is another name for disjunction as the addition is involved in this gate. The “v” symbol denotes the disjunction operation of binary algebra and is written as P v Q. Let us have two digits such as P & Q then the sum operation is applied with the help of the truth table.

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The number of rows is decided by the total number given in the expression, as P & Q are two numbers then the number of rows is 22 = 4.

PQP v Q
11
11
111
  • Negation

Negation is the transpose of the binary variables such as if the value is true the negation will make it false and if the value is false the negation make it true. Such as if the term is 1 it would be transformed into 0 and 0 will be transformed into 1. 

The negation operation is widely used to reverse the process. NOT is another name for negation. 

InputOutput
1
1

How do calculate the problems of Binary algebra?

The problems of binary algebra can be solved with the help of the following two ways.

  1. By using a Boolean calculator
  2. Manually

Here is a brief introduction to the above methods. 

  • By using calculator

The problems of binary algebra can be solved with the help of a Boolean algebra calculator. This calculator will provide the result in two ways such as by using a truth table and by using theorems.  

Here are the steps to using a binary algebra calculator.

  • Enter the expression
  • Hit the calculate button.
  • The solution with steps will come in a couple of seconds. 
  • Manually

Example 1

Calculate the binary algebra of the given function. 

[(P + R) * Q] * [(P * Q) + R]

Solution 

Step 1: Write the given Boolean expression.

[(P + R) * Q] * [(P * Q) + R]

Step 2: Identity how many rows will be involved in the truth table with the help of given digits.

Term in the given expression = n = 3

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According to formula 

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

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

Step 3: According to the given terms make the truth table for three variables.

PQRP + RP * Q(P + R) * Q(P * Q) + R[(P + R) * Q] * [(P * Q) + R]
111
1
111111
11
1111
1111111
11111111

Alternately

The above expression of Boolean algebra can also be solved with the help of theorems. 

Step 1: First of all, use the sum of products rule and write the given expression according to it.

[(P + R) * Q] * [(P * Q) + R] = Q(P + R)(QP + R)

[(P + R) * Q] * [(P * Q) + R] = QPPQ + QPRQ + PRQ + RRQ

Step 2: Now apply the idempotent rule multiplication such as AA = A

[(P + R) * Q] * [(P * Q) + R] = RQR + RQP + QPQ + RQPQ

Step 3: Now apply the idempotent rule addition such as A + A = A

[(P + R) * Q] * [(P * Q) + R] = PQQR + PQQ + QR + QPR

[(P + R) * Q] * [(P * Q) + R] = PRQ + QR + RQP + QQP

Step 4: Now make factors.

[(P + R) * Q] * [(P * Q) + R] = QPQ + QPR + QR

Step 5: By identity law A + 1 = 1

[(P + R) * Q] * [(P * Q) + R] = QR + PR +PRQ

[(P + R) * Q] * [(P * Q) + R] = QR (P + 1) + QP (R + 1)

[(P + R) * Q] * [(P * Q) + R] = R (1) Q + P (R + 1) Q

[(P + R) * Q] * [(P * Q) + R] = RQ + (R + 1) QP

[(P + R) * Q] * [(P * Q) + R] = RQ + (1) QP

[(P + R) * Q] * [(P * Q) + R] = RQ + QP

Final Words

In this article, we have explained all the basics of binary algebra along with its definition, operation, and examples. We have mentioned how to calculate the problems of Boolean algebra by using truth tables and theorems.