What is boolean algebra Its laws and rules with circuit diagram

1. Boolean algebra
Computer organization & architecture

2. Content
 What is boolean algebra
 Laws of boolean algebra
 Rules in boolean algebra
 De morgan’s law

3. What is boolean algebra
 Boolean algebra is a division of mathematics that deals
with operations on logical values and incorporates
binary variables.
 Boolean Algebra is used to analyze and simplify the
digital (logic) circuits. It uses only the binary numbers
i.e. 0 and 1. It is also called as Binary Algebra or
logical Algebra. Boolean algebra was invented by
George Boole in 1854.
 The distinguishing factor of Boolean algebra is that it
deals only with the study of binary variables. Most
commonly Boolean variables are presented with the
possible values of 1 ("true") or 0 ("false"). Variables can
also have more complex interpretations, such as in set
theory. Boolean algebra is also known as binary
algebra.

4. Commutative Law
 This law states that no matter in which order we use the
variables. It means that the order of variables doesn't matter.
In Boolean algebra, the OR and the addition operations are
similar. In the below diagram, the OR gate display that the
order of the input variables does not matter at all.
 For two variables, the commutative law of addition is written
as:
 A+B = B+A
 For two variables, the commutative law of multiplication is
written as:
 A.B = B.A

5. Associative Law
 This law states that the operation can be performed in any order
when the variables priority is same. As '*' and '/' have same priority.
In the below diagram, the associative law is applied to the 2-input
OR gate.
 For three variables, the associative law of addition is written as:
 A + (B + C) = (A + B) + C
 For three variables, the associative law of multiplication is written as:
 A(BC) = (AB)C

6. Distributive Law:
 According to this law, if we perform the OR operation of
two or more variables and then perform the AND
operation of the result with a single variable, then the
result will be similar to performing the AND operation of
that single variable with each two or more variable and
then perform the OR operation of that product. This law
explains the process of factoring.
 For three variables, the distributive law is written as:
 A(B + C) = AB + AC

7. De Morgan’s Theorem
 There are two “de Morgan’s” rules or theorems,
 According to the first theorem, the complement result of the AND operation is equal to the OR operation of
the complement of that variable. Thus, it is equivalent to the NAND function and is a negative-OR function
proving that (A.B)' = A'+B'
 According to the second theorem, the complement result of the OR operation is equal to the AND operation of
the complement of that variable. Thus, it is the equivalent of the NOR function and is a negative-AND function
proving that (A+B)' = A'.B'

8. Rules of Boolean algebra