Basics of Boolean Algebra in Digital Electronics

In the time middle of the 19th century, mathematician George Boole introduced a new form of mathematical reasoning system which is widely known as Boolean Algebra. This is a special branch of mathematics in which all logic is based on only two representations True(1) and False,(0). In modern science, Boolean algebra has become a very integral part of applications in different fields like –> computer science, digital electronics, and mathematical logic reasoning.

Table of Content

• Definition of Boolean Algebra
• Examples of Boolean Algebra
• Understanding Boolean Algebra through Truth Table
• Advantages of Boolean Algebra
• Disadvantages of Boolean Algebra

Definition of Boolean Algebra

Boolean algebra is a special mathematical way to express relations (logics) between variables. Mainly there are three basic logic gates or operators which are AND(*), OR(+), and NOT(‘). By using only these three operators we can develop any complex Boolean expressions and can determine the truth values of the variables which are present in that particular expression. In Boolean algebra, every operator is must be a binary variable i.e. they can either take a true(1) or false(0) value.

Examples of Boolean Algebra

Let’s solve a Boolean expression:

(A AND B) OR (C AND (NOT D)), given A=B=1, C = 0, D = 0

Let’s replace the expression with given values part by part.

• (A AND B) = (1 AND 1) = 1
• (C AND (NOT D)) = (0 AND (NOT 0)) = (0 AND 1) = 0
• (A AND B) OR (C AND (NOT D)) = (1 OR 0) = 1

So, the final answer for the given Boolean expression is 1.

Understanding Boolean Algebra through Truth Table

Let’s understand it using truth table:

Let Y= (A AND B) OR (C AND (NOT D))

Input				Output
A	B	C	D	A AND B	C AND (NOT D)	Y
0	0	0	0	0	0	0
0	0	0	1	0	0	0
0	0	1	0	0	1	1
0	0	1	1	0	0	0
0	1	0	0	0	0	0
0	1	0	1	0	0	0
0	1	1	0	0	1	1
0	1	1	1	0	0	0
1	0	0	0	0	0	0
1	0	0	1	0	0	0
1	0	1	0	0	1	1
1	0	1	1	0	0	0
1	1	0	0	1	0	1
1	1	0	1	1	0	1
1	1	1	0	1	1	1
1	1	1	1	1	0	1

Logical Circuit Diagram of Boolean Algebra Example

Let’s see it’s circuit diagram

Logical Circuit Diagram for the given example

Advantages of Boolean Algebra

• Simplification: By using Boolean algebra we can decompose a complex logic to its basic simple form without changing the resultant value. So, it minimizes the logic circuit complexity which reduces the number of required logic gates and cost.
• Formalism: We can analyze logic relationships by using Boolean algebra. This allows analyzing logical relationships which helps to accurate model designing and manipulation of logical operations.
• Standardization: Boolean Algebra ensures consistency and uniformity across different fields and applications.

Disadvantages of Boolean Algebra

• Complexity in Real-World Problems: Boolean algebra only takes two values(0 & 1) but in real-world scenario many applications requires multi-valued or fractional valued or higher order values.
• Steep Learning Curve: To implement Boolean algebra an extensive training is required as it has very unique rules, theorems and abstract nature.

Applications of Boolean Algebra

For example we will only see the brief details about the basic gates and solve one complex Boolean expression.

• AND operator: This operator is represented with ( * specifically with ⊗). In this case if both the variables have true(1) value then only result will be true(1), otherwise result will be false(0).
• OR operator: This operator is represented with (+, specifically ⊕). In this case if only one variable has true(1) value then the result will be true(1). If and only if both variable have false(0) value then the result will be false(0).
• NOT operator: This operator is represented with (‘). Here, if the variable has true(1) value then the result will be false(0) and vice versa.

FAQs on Basics of Boolean Algebra

Q.1: What do you mean by Boolean Algebra and why it has become essential in Computer Science and Electronic field?

Answer:

Boolean Algebra is a special branch of Mathematics. But here only binary operations can be performed. Boolean algebra handles logical operations on binary variables and provides output only in the terms of true(1) and false(0).

Boolean algebra has become essential part in the field of Computer Science and Electronics because by using this special mathematics we can design complex digital circuits, perform logical operations in programming language and perform complex queries in database and network routing.

Q.2: List some practical applications of Boolean Algebra.

Answer:

Boolean algebra is used by various real-world applications which are listed below–>

1. Search Engines use Boolean algebra for natural language processing to provide results relevant to the user queries.

2. Boolean algebra is used to develop complex search queries in legal databases.

3. Boolean algebra has very effective decision-taking and management skills which can be used to develop strong business rules and handle critical conditional scenarios faced by industries.

Q.3: What are the limitations of Boolean Algebra?

Answer:

Though Boolean algebra is an essential portion of Computer Science and Electronics field, still it has several limitation which are listed below:

1. If a system needs to access continuous data or probabilistic data then Boolean algebra can’t be used as it has only two value(0, 1). It can only handle binary data or discrete use of data.

2. The simplification process of Boolean expression is very time-consuming and lengthy which may take high CPU resources and results higher cost.

3. For complex real-time systems, Boolean algebra is not efficient as there is a use of multivalued or higher order logics.

Q.4: Define truth tables and their relation with Boolean Algebra.

Answer:

Truth tables can be defined as a tabular format representation of all possible combinations of input values present in logical expression along with the corresponding output of each combination. The rows of the truth tables consisting of a specific combination of input sets and the final column holds the corresponding output values(0 or 1). The output will directly dependent on the Boolean expression. So, adversely we can say that Boolean Algebra and truth tables are related to each other as truth tables evaluate the validity of a Boolean expression for different possible input values. In modern technology where complex logic circuit design is involved, truth tables are playing an essential role to verify(cross-check) the aligned output with desired output.

Q.5: How Boolean algebra can be used in complex decision-making processes?

Answer:

Boolean algebra can be used very efficiently in complex decision-making processes by decomposing complex expressions into simple logic(resultant output will be same). Complex decision-making processes are nothing but some logical expressions but there is a presence of various conditions and choices. We can handle these conditions using multiple basic operators like AND, OR and NOT. We can organize the multiple number choices by using truth table. So, that Boolean algebra can be employed in various critical decision-making fields like business models, engineering applications etc.

Q.6: Write about the process of simplifying a Boolean expression.

Answer:

To solve or simplify a complex Boolean expression we need to know about Boolean laws. Like Complement law, Idempotent Law and Commutative law.

In Complement law, if a variable AND-ed with it’s compliment then it will be 0(A.A”=0) and if OR-ed then it will be 1(A+A’=1).

In Idempotent Law, If a variable is AND-ed or OR-ed with itself then for both cases the answer will be the variable itself.

In Commutative law, the order of AND-ing or OR-ing will make no difference(A+B=B+A, A.B=B.A).

Also there are de Morgan’s law, Boolean postulates which we need to solve it.