Laws of Boolean Algebra
Boolean algebra, developed by the English mathematician George Boole in the
19th century, is a mathematical system that deals with logical operations on
Boolean variables, which can only take two values: 0 (false) and 1 (true).
The laws of Boolean algebra define the rules that govern Boolean operations
and enable the manipulation and simplification of Boolean expressions.
As we saw in the previous article,
*Boolean Algebra and Simplification of Algebraic Expressions*, Boolean algebra plays a crucial role in digital electronics. It allows us
to model the behavior of logic circuits, which are the fundamental building
blocks of computers and other digital systems. In this article, we will
explore the fundamental laws of Boolean algebra and discover its practical
applications in digital electronics.
Commutative Law in Boolean Algebra
The commutative law in Boolean algebra states that the order of variables in a Boolean expression does not affect the result of the operation. In other words, variables can be swapped without changing the value of the expression.
For AND: A · B = B · A
For OR: A + B = B + A
For XOR: A ⨁ B = B ⨁ A
This rule can be generalized to larger expressions with multiple variables grouped within parentheses. Here are some examples:
For AND: (A · B · C) = (B · A · C) = (C · B · A)
For OR: (A + B + C) = (B + A + C) = (C + B + A)
For AND: (A+B) · (C+D) = (C+D).(A+B)
For OR: A.B + C.D =C.D+ A.B
These examples show that the order of variables in AND or OR operations can be rearranged without affecting the result of the Boolean expression
Associative Law in Boolean Algebra
The associative law in Boolean algebra states that the grouping of variables in a Boolean expression does not affect the result of the operation. In other words, variables can be grouped in different ways without changing the value of the expression.
For AND: (A · B) · C = A · (B · C)
For OR: (A + B) + C = A + (B + C)
he rule can be generalized to larger and more complex Boolean expressions with multiple variables.
For example:
For AND: (A · B) · (C · D) = A · (B · C) · D
For OR: (A + B) + (C + D) = A + (B + C) + D
This shows that the grouping of variables in AND or OR operations can be rearranged in any way without affecting the outcome of the Boolean expression.
Distributive Law in Boolean Algebra
The distributive law is the foundation of many theorems and identities in Boolean algebra, which are used to analyze and manipulate Boolean expressions.
The distributive law in Boolean algebra is similar to that in ordinary algebra. It allows a Boolean expression to be broken down by distributing a common factor. In other words, it enables the transformation of a logical conjunction (AND) into multiple disjunctions (OR), or vice versa.
For AND over OR: A · (B + C) = (A · B) + (A · C)
For OR over AND: A + (B · C) = (A + B) · (A + C)
Yes, the distributive law can also be applied to more complex Boolean expressions with multiple variables.
For example:
For AND over OR: A · (B + C + D) = (A · B) + (A · C) + (A · D)
For OR over AND: A + (B · C · D) = (A + B) · (A + C) · (A + D)
This demonstrates that the distributive property can be extended to handle intricate expressions with many variables, allowing for flexible manipulation and simplification in Boolean algebra.
Rules of Boolean Algebra
Using the laws and theorems introduced in the previous articles, any Boolean expression can be simplified to its most basic form. However, when working with Boolean algebra, certain recurring expressions often appear. These expressions can be treated as general rules that can be applied directly, but it’s important to understand where they come from.
- A+A.B=A
- A.(A+B)=A
- AB+AB=A
- (A+B).(A+B)=A
- A+AB=A+B
- A.(A+B)=A+B
- AB+AC+BC=AB+AC
- (A+B).(A+C).(B+C)=(A+B).(A+C)
De Morgan's Laws
Despite all the laws and rules we have covered so far, it is not always possible to simplify certain Boolean expressions to their simplest form. This is where De Morgan's laws come into play, providing a significant contribution to Boolean algebra by offering a powerful tool for further simplifying complex expressions.
De Morgan's laws play a crucial role in Boolean algebra by establishing a fundamental connection between the logical operators "AND" (conjunction) and "OR" (disjunction), and by providing effective tools for simplifying expressions.
De Morgan's laws are not limited to simplifying Boolean expressions by grouping negative terms. They are also highly useful for breaking down extended negations applied to complex expressions.
The laws are stated as follows:
First Law: The negation of a conjunction is equivalent to the disjunction of the negations:
A.B=A+B
Second Law: The negation of a disjunction is equivalent to the conjunction of the negations:
A+B=A.B
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