- In 1854 George Boole developed Boolean algebra
- Switching algebra was introduced by C.E. Shannon

**Huntington postulates:**

1. A. Closure with respect to ‘+’

B. Closure with respect to
‘.’

__Note__: a set is said to be closed with respect to a operator if, for every pair of elements of s, the result of operation is also a unique element of S. For example N = {1, 2, 3…} is closure with respect to operator ‘+’ because 2+4=6 is also element of N.

a + b = b + a

a . b = b . a

3. A. identity element with respect to ‘.’ is ‘1’

a . 1 = 1. a = a

B. identity element with
respect to ‘+’ is ‘0’

a + 0 = 0 + a = a

4. Distributive over + and .

a + ( b . c ) = ( a + b ) .
( a + c ) with respect to +

a . ( b + c ) = ( a . b ) +
( a . c ) with respect to .

5. For every element ‘a’ there exists a unique element called a’

a+a’ = 1

a.a’=0

__Note:__Huntington postulates doesn’t include associative law

Associative
with respect to + and .

a + ( b + c )
= ( a + b ) + c

a . ( b . c )
= (a . b ) . c

**Differences between normal and Boolean algebra:**

- Complement of a number doesn’t exist in normal algebra
- Distributive law over + is valid for Boolean algebra but not normal algebra
- Boolean algebra doesn’t have additive and multiplicative inverse

**Duality:**

- Dual of an expression can be obtained by replacing all ‘+’ with ‘.’, all ‘0’ with ‘1’, all ‘1’ with ‘0’, all ‘.’ With ‘+’.
- If Boolean expression is valid it’s dual is also valid
- Example:

(a+b).c=(a.c)+(b.c) – is dual of above expression

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