The document discusses Boolean algebra, which uses binary numbers (0 and 1) to analyze and simplify digital logic circuits. It was invented by George Boole in 1854. The document outlines several important rules of Boolean algebra, including commutative, associative, distributive, identity, idempotent, complement, and double negation laws. It also discusses de Morgan's theorem and finding the dual of Boolean expressions.

1. Prof. Neeraj Bhargava
Pooja Dixit
Department of Computer Science, School of Engineering & System Sciences
MDS University Ajmer, Rajasthan

2.  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.
Rule in Boolean Algebra
 Following are the important rules used in Boolean algebra.
 Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
 Complement of a variable is represented by an overbar (-). Thus, complement of
variable B is represented as . Thus if B = 0 then = 1 and B = 1 then = 0.
 ORing of the variables is represented by a plus (+) sign between them. For example
ORing of A, B, C is represented as A + B + C.
 Logical ANDing of the two or more variable is represented by writing a dot
between them such as A.B.C. Sometime the dot may be omitted like ABC.

3.  Boolean Laws/ Properties of Boolean algebra
There are seven types of Boolean Laws.
1. Commutative law: In a group of variables connected by operators AND or OR,
the order of the variables does not matter.
 Boolean addition (OR): A+B = B+A
 Boolean multiplication (AND):A•B = B•A
Commutative law states that changing the sequence of the variables does not
have any effect on the output of a logic circuit.
2. Associative law:
 This law states that the order in which the logic operations are performed is
irrelevant as their effect is the same.
 Boolean addition (OR): (A+B)+C = A+(B+C) = A+B+C
 Boolean Multiplication (AND): (A•B)•C = A•(B•C) = A•B•C = ABC

4. 3. Distributive Laws
 The same answer is arrived at when multiplying (ANDing) a variable by a group of
bracketed variables added (ORed) together, as when each multiplication (AND) is
performed separately.
 Law a is similar to factoring in normal algebra, but law b is unique to Boolean algebra
because unlike normal algebra, where A x A=A2, in Boolean algebra A•A = A
 a.) A•(B+C) = A•B+A•C
 b.) A+(B•C) = (A+B) • (A+C)
4. Identity Elements
 When the variable is AND with 1 and OR with 0, the variable remains the same, i.e.,
a. A•1 = A
 b. A+0 = A
 C. A•0=0
 D. A+1=1

5. 5. Idempotent Law
 When the variable is AND and OR with itself, the variable remains same or
unchanged, i.e.,
 B.B = B
 B+B = B
6. Complementarily Law: A term AND´ed with its complement equals “0” and a
term OR´ed with its complement equals “1”
 A + A’ = 1
 A⋅A’=0
 A+B = A.B
 A.B = A+B
 A . A’ = 0 A variable AND’ed with its complement is always equal to 0
 A + A’` = 1 A variable OR’ed with its complement is always equal to 1

7. Double Negation Law : A term that is inverted twice is equal to the original term
(x’)’ = x
de Morgan’sTheorem

6.  It change positive into negative that is state 1
change into state 0 and also ‘+’ sign is replaced by
‘.’ dot operator.

8.  Find the dual of boolean expression:
 X(y+z)=x y + x z
 X+(yz)
 (X+y). (x+z)