The document discusses Boolean algebra laws and theorems that are used to simplify Boolean expressions and logic. Some of the key laws and theorems covered include the associative, distributive, commutative, absorption, and duality laws. De Morgan's theorems are also explained, which relate to taking the complement of a sum or product of variables. Truth tables are used to demonstrate De Morgan's theorems. The overall purpose is to provide some of the fundamental laws and theorems of Boolean algebra that can be applied to simplify logical expressions.

1. Boolean Algebra Laws and
Theorems
Boolean Algebra is a form of mathematical
algebra that is used in digital logic in digital
electronics. Albebra consists of symbolic
representation of a statement (generally
mathematical statements). Similarly, there are
expressions, equations and functions in Boolean
algebra as well.

2. Boolean Algebra Laws and Theorems
• The main aim of any logic design is to simplify the
logic as much as possible so that the final
implementation will become easy. In order to simplify
the logic, the Boolean equations and expressions
representing that logic must be simplified.
• So, to simplify the Boolean equations and expression,
there are some laws and theorems proposed. Using
these laws and theorems, it becomes very easy to
simplify or reduce the logical complexities of any
Boolean expression or function.

3. The article demonstrates some of the most commonly
used laws and theorem is Boolean algebra.
• Basic Laws and Proofs
The basic rules and laws of Boolean algebraic
system are known as “Laws of Boolean algebra”.
Some of the basic laws (rules) of the Boolean
algebra are
• i.Associative law
• ii. Distributive law
• iii. Commutative law
• iv. Absorption law
• v. Consensus law

4. 1.Associative Law
Associate Law of Addition
Statement:
• Associative law of addition states that OR ing more than
two variables i.e. mathematical addition operation
performed on variables will return the same value
irrespective of the grouping of variables in an equation.
It involves in swapping of variables in groups.
The Associative law using OR operator can be
written as
• A+(B+C) = (A+B)+C = A + (B + C) = B + (C + A)
Eg:(0 + 1) + 0 = 0 + (1 + 0)
1 + 0 = 0 + 1
1 = 1

5. • Associate Law of Multiplication
Statement:
Associative law of multiplication states that
ANDing more than two variables i.e.
mathematical multiplication operation
performed on variables will return the same
value irrespective of the grouping of variables
in an equation.
• The Associative law using AND operator can
be written as
• A * (B * C) = (A * B) * C

6. • Distributive law
This is the most used and most important law in Boolean
algebra, which involves in 2 operators: AND, OR.
Statement:
• The multiplication of two variables and adding the result
with a variable will result in same value as multiplication
of addition of the variable with individual variables.
• In other words, ANDing two variables and ORing the
result with another variable is equal to AND of ORing of
the variable with the two individual variables.
• Distributive law can be written as
A + BC = (A + B)(A + C)
similarly,
A (B+C) = (A B) + (A C)

7. • Commutative law
Statement:
• Commutative law states that the inter-
changing of the order of operands in a
Boolean equation does not change its result.
• Using OR operator → A + B = B + A
• Using AND operator → A * B = B * A
• This law is also has more priority in Boolean
algebra.
Eg:1 + 0 = 0 + 1, 1 * 0 = 0 * 1

8. • Absorption Law
Absorption law involves in linking of a pair of
binary operations.
A+A=A
A.A=A
A+AB = A
A(A+B) = A
A+ĀB = A+B
A.(Ā+B) = AB
5th and 6th laws are also called as Redundancy laws.

9. 1.Statement : A + A = A
LHS=A+A
=(A+A).1
=(A+A).(A+Ā)
=A+(A. Ā)
=A(RHS)
2.Statement : A . A = A
LHS=(A.A)+0
=(A.A) +(A. Ā)
=A.(A+Ā)
=A+(1)
=A(RHS)

10. 3.Statement : A + AB = A
Proof:
A + AB = A.1 + AB → since A.1 = A
=A(1+B) → since 1 + B = 1
= A.1
= A
4.Statement :A (A + B) = A
Proof:
A (A + B) = A.A + A.B
= A+AB → since A . A = A
= A (1 + B)
= A.1
= A

11. 5.Statement :A + ĀB = A + B
Proof:
A + ĀB = (A + Ā) (A + B) → since A+BC
=(A+B)(A+C) using distributive law
= 1 * (A + B) → since A + Ā = 1
=A + B
6.Statement : A * (Ā+B) = AB
Proof: A * (Ā + B) = A. Ā + AB
= AB → since A Ā = 0

12. Duality Principle in Boolean algebra
• Statement:
Duality principle states that “The Dual of the expression can
be achieved by replacing the AND operator with OR
operator, along with replacing the binary variables, such as
replacing 1 with 0 and replacing 0 with 1”.
(or)
One expression can be obtain from another expression by
replacing the every 1 with 0, every 0 with 1, every (+) with
(.), every (.) with (+). Any pair of expression satisfying this
property is called dual expression. This characteristic of
Boolean algebra is called the principle of duality.
Eg: A + B = 0,
A * B = 1.

13. De Morgan’s Theorem:
Statement 1:“The compliment of the product of
2 variables is equal to the sum of the
compliments of individual variables”.
(A.B)’ = A’ + B’
Statement 2: “The compliment of the sum of
two variables is equal to the product of the
compliment of each variable”.
(A + B)’ = A’.B’

14. Truth Tables
The De Morgan’s laws are simply explained by using the truth tables.
The truth table for De Morgan’s first statement((A.B)’ = A’ + B’) is given below.
The truth table for De Morgan’s second statement ((A + B)’ = A’.B’) is given
below.