BOOLEAN ALGEBRA: -
It was introduced by George Boole in (1847).
It is a mathematical system that deals with the logical operations and values. Unlike ordinary algebra that has numbers, it uses Binary values (0, 1) and Logical operators (AND, OR, NOT). It provides theoretical foundation for digital circuit design.
BOOLEAN LAWS: -
1.) Idempotent law: -
a . a = a
a + a = a
⚡ Repetaion of same variable dosent change resut
2.) Associative law:-
P + (Q + R) = (P + Q) + R
P ⋅ (Q ⋅ R) = (P ⋅ Q) ⋅ R
⚡ Grouping doent matter
3.) Distributive law:-
P + (Q ⋅ R) = (P + Q) ⋅ (P + R)
P ⋅ (Q + R) = (P ⋅ Q) + (P ⋅ R)
⚡ Distribution over AND/OR
4.) Comutative law:-
P + Q = Q + P
P ⋅ Q = Q ⋅ P
⚡ Order dosent matter
5.) De’ Morgans law:-
(P + Q)′ = P′ ⋅ Q′
(P ⋅ Q)′ = P′ + Q′
⚡ Negation over AND/OR flips operation and complement
6.) Identity law:-
P + 0 = P
P ⋅ 1 = P
⚡ Multiplying 1 and Adding 0 dosent change result
7.) Involution law (Double Negation)
(P′)′ = P
8.) Complement law:-
P + P′ = 1
P ⋅ P′ = 0
⚡ P and its complement always give fixed results
9.) Absorption law:-
P + (P ⋅ Q) = P
P ⋅ (P + Q) = P
⚡ Redundant terms can be observed
🚐 Minimizing Boolean Expressions
Boolean algebra is a method where we use special rules (like distributive, associative, and complement laws) to simplify Boolean expressions.
In simple words, we use these rules to make long logic expressions shorter and remove extra or useless terms.
Example: Simplify the Boolean function F = AB + (AC)′ + AB′C(AB + C)
Solution: F
= AB + (AC)′ + AB′C(AB + C)
= AB + A′ + C′+ AB′C.AB + AB′C.C
| IN ( AB′C.AB ) A × A → A C → C But look at ( B′ × B ) B′ × B = 0 |
IN ( AB′C.C ) = A × B′ × C × C C × C = C So: AB’C·C = AB′C |
= AB + A′ + C′ + 0 + AB′C
AB = AB(C + C′) Because (C + C′ = 1) So: AB = ABC + ABC′ |
= ABC + ABC′ + A′ + C′ + AB′C
ABC + ABC′ Take AC common:
= AC(B + B′) |
We know: B + B′ = 1 So:
AC × 1 = AC |
= AC + A′ + C′ + ABC'
C' + ABC' = C' ( 1 + AB ) Where: 1 + AB = 1 So: C' ( 1 + AB ) = C' |
= AC + C′+A′
C′+A′ Applying De' Morgan law C′+A′ = (AC)′ |
=
AC + (AC)′ = 1
∴ THE EXPRESSION IS MINIMIZED
