# Boolean Algebra

In this tutorial, you will learn about the basics of Boolean Algebra. You will learn about the fundamental laws of Boolean Algebra, AND, OR and NOT operations, their properties, and De-Morgan’s law.

Contents:

## What is Boolean Algebra?

Boolean Algebra is the technique to work with multivariable logical expressions which either result in true or false outputs expressed as 1 or 0 respectively. It explicitly focuses on logical operations and is extensively used in digital subjects. There are three fundamental operations in Boolean algebra given as: –

• Conjunction (AND)
• Disjunction (OR)
• Negation (NOT)

Here are some points related to Boolean algebra: –

• Variables in Boolean algebra are mostly represented with capital letters. For example, A, B, C, etc. The dependent variable is shown in small letters. Any variable used in its complemented form is shown with a bar sign on top of it or an apostrophe.
• Any function expressed in Boolean algebra is made up of these three operations only. All higher operations are derived from these three operations.
• Any of the variables used in a function can have only two values either 1 (HIGH) or 0 (LOW).

Examples of Boolean function: –
f (A, B, C) = AB + BCA’+ CA, f (X, Y) = X’Y + Y’X

## Fundamental Laws of Boolean Algebra

Here are a few fundamental laws and identities in Boolean Algebra that are used in all the operations.

1. Commutative Law: – According to the commutative law, the order of two variables in a function does not matter. An operation is said to be commutative if it gives the same result on reversing the order of two variables and performing the same operation.
2. Associative law: – According to the associative law, in the matter of three variable functions, performing the operation on the first two and then the third is the same as operating first on the last two variables and then on the first variable.
3. Distributive law: – According to the distributive law, if an operation can be distributed over two variables on opening brackets and the result remains unchanged as it was without opening brackets, it is said to be distributive.
4. Multiplicative Identity: – A value is said to be a multiplicative identity if on multiplying it with a variable, we get the same variable.
5. Additive Identity: – A value is said to be an additive identity if by adding it to the variable, we get the same variable.

## Conjunction (AND Operation)

The AND operation is equivalent to the intersection logic used in sets of mathematics. For any number of variables, an AND operation returns the result as true only when all the variables are true. Even if one value is false, the entire function becomes false.

It is shown by the dot product. For example, A and B can be written as A.B or AB simply.
Its truth table is given as: –

A B A and B
0
0
0
0
1
0
1
0
0
1
1
1

As shown in the table, the AND operation gives a value of 1 only when both the input values are 1. For example, if f=ABC, the function f will result in 1 only when all three A, B, C are 1. The AND operation can be extended to any number of variables.

## Properties of AND Operation

Here are the properties of AND operation listed.

• It is commutative.
AB = BA
• It is associative.
(AB)C = A(BC)
• It is distributive over addition.
A(B+C) = AB + AC
• Any variable ANDed with itself gives the same variable.
A.A = A
• Any variable ANDed with zero gives zero.
A.0 = 0
• The multiplicative identity in AND operation is one.
A.1 = A

## Disjunction (OR Operation)

The OR operation is equivalent to the union logic used in sets of mathematics. For any number of variables, the OR operation returns the result as true if any one of the values is true (1). If both values are false only then the entire function becomes false.

It is shown by the plus sign. For example, A and B can be written as A + B.
Its truth table is given as: –

A B A or B
0
0
0
0
1
1
1
0
1
1
1
1

As shown in the table, the result of OR operation is 0 only when both A and B are 0.

## Properties of OR Operation

Here are the properties of OR operation listed.

• It is commutative.
A + B = B + A
• It is associative.
(A + B) + C = A + (B + C)
• It is distributive over multiplication.
A + BC = (A + B) (A + C)
• Any variable ANDed with itself gives the same variable.
A + A = A
• Any variable ORed with zero will give the same variable.
A + 0 = A
• Any variable ORed with 1 will give 1.
A + 1 = 1
• The additive identity in OR operation is zero.

## Negation (NOT Operation)

In the negation operation, we use complemented value of an expression. It is represented with a long bar on the top of a Boolean expression. It is also represented with a single apostrophe. For example, A’, (A+BC)’ etc.

Here is the truth table for negation.

A A’
0
1
1
0

Here are the properties of negation listed: –

• Negation of negation returns the same variable.
(A’)’ = A
• The value of A ORed with A’s complement is always one.
A + A’ = 1
• The value of A ANDed with A’s complement is always zero.
A . A’ = 0

## De-Morgan’s Law

According to De-Morgan’s law,

• If we complement the addition between two variables (or operation), the resultant is the product of the two in their complemented forms individually.
(A + B)’ = A’B’
• If we apply complement to the product of two variables (and operation), the resultant is an OR operation between the complemented forms of the two variables.
(AB)’ = A’ + B’

De-Morgan’s Law is extensively used in Boolean Algebra to interconvert expressions. De-Morgan’s law can be applied to more than two variables also. In that case, all the n variables will get complemented with the operation reversed among them.

## Key Points to Remember

Here are the key points to remember in “Boolean Algebra”.

• Boolean Algebra has the fundamental operations of AND, OR, and NOT.
• The AND operation between any number of variables returns true as the value only if all the values are true individually.
• The OR operation between any number of variables returns true if any of the values are true.
• The NOT operation when applied to an expression returns the complemented form of the expression.
• The AND operation is analogous to the intersection operation of sets whereas the OR operation is analogous to the union operation of sets.
• AND as well as OR operations are commutative, associative, and distributive.
• The De-Morgan’s laws are:
(A + B)’ = A’B’ and (A.B)’ = A’ + B’

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