Laws of Boolean Algebra

Boolean Algebra refers to the form of mathematical algebra that we use in the case of digital logic in digital electronics. Algebra consists of the symbolic representation of any statement, and these statements are generally mathematical. Similarly, in Boolean algebra, there are equations, expressions, and functions as well.

Various laws of Boolean algebra allow us to simplify very complex formulas into some expressions that are more manageable. In this article, we will take a look at the Laws of Boolean Algebra according to the GATE Syllabus for CSE (Computer Science Engineering). Read ahead to learn more.

Table of Contents

Laws of Boolean Algebra
- Properties of Boolean Algebra
- Boolean Postulates

Properties of Boolean Algebra

The switching algebra is also called Boolean Algebra. Basically, it is used to analyse digital gates along with the circuits. It is logical to perform a mathematical operation on the binary numbers, namely on ‘1’ and ‘0’. Boolean Algebra consists of various basic operators such as AND, OR, NOT etc. The given operations are represented using the ‘.’ for AND along with ‘+’ for OR. We can perform the operations on variables that are represented using capital letters, for example, ‘A’, ‘B’, ‘C’, etc.

Properties of Switching Algebra

The primary aim of a logic design is the simplification of the logic as much as possible. We do this so that the final implementation becomes comparatively easy. To simplify the logic, all the Boolean equations along with expressions that represent the logic have to be simplified.

Here are a few laws that define the various properties present in a switching algebra:

Annulment Law

It states that a variable that has been ANDed with 0 gives us a 0, while a variable that has been ORed with 1 gives us a 1, i.e.,

X.0 = 0
X + 1 = 1

Identity Law

This law states that a variable would remain unchanged when it is ANDed with ‘1’ or ORed with ‘0’, i.e.,

X.1 = X
X + 0 = X

Idempotent Law

This law states that a variable would remain unchanged when it is ANDed or ORed with itself, i.e.,

X + X = X
X.X = X

Complement Law

This law states that in case a complement is added to any variable, then it would give one, whereas when we multiply this variable with its own complement, then it would result in ‘0’, i.e.,

X + X’ = 1
X.X’ = 0

Double negation Law

This law states that whenever a variable is with two negations, then its symbol would ultimately get cancelled out while the original variable is obtained with it, i.e.,

((X)’)’ = X

Commutative Law

The order of the variable or two different terms does not matter according to this law. It can be represented as follows,

X + Y = Y + X
X.Y = Y.X

Associative Law

According to this law, the order of an operation does not matter when the priority of the given variables are similar, such as ‘*’ and ‘/’, i.e.,

X + (Y + Z) = (X + Y) + Z
X.(Y.Z) = (X.Y).Z

Distributive Law

Using this law, we try to understand how the opening up of the brackets available in an operation, i.e.,

X.(Y+Z) = (X.Y) + (X.Z)
X + (Y.Z) = (X + Y).(X + Z)

Absorption Law

Using this law, we absorb similar variables, i.e.,

X.(X + Y) = X
X + XY = X

De Morgan Law

This law states that the operation of an OR or an AND logic circuit stays unchanged whenever all inputs are inverted, while the operator actually changes from AND to OR whenever the outputs are inverted, i.e.,

(X.Y)’ = X’ + Y’
(X + Y)’ = X’.Y’

Boolean Postulates

While not using Boolean Laws in their own right, here are a few Mathematical Laws that we can use for simplifying various Boolean Expressions.

0 . 0 = 0 0 AND’ed with its own self would be equal to 0
1 . 1 = 1 1 AND’ed with its own self would be equal to 1
1 . 0 = 0 1 AND’ed with 0 would be equal to 0
0 + 0 = 0 0 OR’ed with its own self would be equal to 0
1 + 1 = 1 1 OR’ed with its own self would be equal to 1
1 + 0 = 1 1 OR’ed with 0 would be equal to 1
1 = 0 The Complement or Inverse of 1 would be equal to 0
0 = 1 The Complement or Inverse of 0 would be equal to 1

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