Properties of Boolean Algebra

The Boolean Algebra uses sets of rules for analyzing digital gates and circuits. In this article, we will be going through the Properties or Laws of the Boolean algebra. So first we will start our article by defining what are the properties of Boolean Algebra, and then we will go through what are Boolean Addition and Multiplication. Then we will go through the different properties of Boolean Algebra such as Annulment, Identity law, Idempotent law, etc.

What are the Properties of Boolean Algebra?

The Boolean Algebra uses sets of rules for analyzing digital gates and circuits which are known as Laws or properties of Boolean Algebra. These laws or properties help to simplify complex Boolean expressions by reducing the number of logic gates required for a given operation.

Variables in the Boolean Algebra are binary which takes only values of either 0 or 1. While expression can be written in an infinite number of variables, each variable can only assume one of these two values. For example, variables labelled as A, B, C, etc., can be utilized to formulate logical expressions such as A + B = C.

Boolean Addition

It is the fundamental operation in Boolean algebra, it is similar to the OR Operation. In digital circuits, it is used to compute the sum term without necessitating the AND operation. The sum term evaluates to true if one or more literals are true, and false if all literals are false. Some of the examples of it are A+B, A'+B, A'+C'.

Boolean Multiplication

It is also one of the fundamental operations in Boolean algebra and it is similar to the AND operation. In a digital circuit, it is used to determine the product term, without using the OR operations. The product term evaluates to true only if all literals are true; otherwise, false. Some of the examples of the AND operation are A.C, A.D.E.

Properties of Boolean Algebra

Switching algebra is also known as Boolean Algebra. It is used to analyze digital gates and circuits It is logical to perform a mathematical operation on binary numbers i.e., on '0' and '1'. Boolean Algebra contains basic operators like AND, OR, and NOT, etc. Operations are represented by '.' for AND, and '+' for OR. Operations can be performed on variables that are represented using capital letters e.g., 'A', 'B' etc. 

Properties of Switching Algebra

The Primary objective of the logic design is to solve the expression to its simplest form. This simplification process is important to ensure that the final implementation of a logic circuit is as Simple as possible. By reducing complexity, we can increase efficiency and ease of implementation and making the overall design process more simple.

Annulment law

a variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1, i.e., 

A.0 = 0 

A + 1 = 1 

Identity law

In this law variable remains unchanged it is ORed with '0' or ANDed with '1', i.e., 

A.1 = A 

A + 0 = A 

Idempotent law

A variable remains unchanged when it is ORed or ANDed with itself, i.e., 

A + A = A 

A.A = A 

Complement law

In this Law if a complement is added to a variable it gives one, if a variable is multiplied with its complement it results in '0', i.e., 

A + A' = 1 

A.A' = 0 

Double Negation Law

A variable with two negations, its symbol gets cancelled out and original variable is obtained, i.e., 

((A)')'=A 

Commutative law

A variable order does not matter in this law, i.e., 

A + B = B + A 

A.B = B.A 

Associative law

The order of operation does not matter if the priority of variables are the same like '*' and '/', i.e., 

A+(B+C) = (A+B)+C 

A.(B.C) = (A.B).C 

Distributive law

This law governs the opening up of brackets, i.e., 

A.(B+C) = (A.B)+(A.C)

(A+B)(A+C) = A + BC 

Absorption law

The absorption law consists of two dual statements:

X.(X+Y)=X

X+XY=X

De Morgan law

In De Morgan law, the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted, i.e., 

(A.B)' = A' + B' 

(A+B)' = A'.B'

Consensus theorem

AB + A'C + BC = AB + A'C

References

1. GATE CS 2013, Question 65
2. GATE CS 2009, Question 6
3. GATE CS 2007, Question 85

Solved Examples

Question 1: Simplify A . B + A . B'.

Solution:

A . B + A . B' = A . (B + B')

= A . (1)

= A

Question 2: Simplify A + A' . B.

Solution:

A + A' . B = (A + A') . (A + B)

= (1) . (A + B)

= A + B

Question 3: Simplify (A + B) . (A + B') + (B . B').

Solution:

(A + B) . (A + B') + (B . B') = (A + B) . (A + B') + (0)

= A + (B . B')

= A + (0)

= A

Question 4: Simplify A . (B + C) + A' . (B + C)

Solution:

A . (B + C) + A' . (B + C) = (A + A') . (B + C)

= 1 . (B + C)

= B + C

Question 5: Simplify A + B . A' + C . C'

Solution:

A + B . A' + C . C' = A + B . A' + 0

= A + B

Properties of Boolean Algebra - Unsolved Problems

• 1. Simplify A + A . B.
• 2. Simplify (A . B) + (A . B') + (B . B').
• 3. Simplify A + B + A' . B.
• 4. Simplify (A . B) + (A . B') + (A' . B).
• 5. Simplify A + B + A . B.
• 6. Simplify (A + B) . (A' + B').
• 7. Simplify A . (B + C) + A' . (B + C).
• 8. Simplify A + B . A' + C . C'.
• 9. Simplify A . B' + A' . B + B . C.
• 10. Simplify A . (B + B').

Related Articles:

• Boolean Algebra
• Boolean Algebraic Theorems

Conclusion

In this article, we have gone through different through what are Properties of Boolean Algebra, then we have gone through Boolean Addition and multiplication in brief and after that we have gone through different laws or properties in detail.