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lets talk about most famous Boolean mathematics..........

The most obvious way to simplify Boolean expressions is to manipulate them in the same way as normal algebraic expressions are manipulated. With regards to logic relations in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns.
A set of rules formulated by the English mathematician George Boole describe certain propositions whose outcome would be either true or false. With regard to digital logic, these rules are used to describe circuits whose state can be either, 1 (true) or 0 (false). In order to fully understand this, the relation between the AND gate, OR gate and NOT gate operations should be appreciated. A number of rules can be derived from these relations as Table 1 demonstrates.

• P1: X = 0 or X = 1
• P2: 0 . 0 = 0
• P3: 1 + 1 = 1
• P4: 0 + 0 = 0
• P5: 1 . 1 = 1
• P6: 1 . 0 = 0 . 1 = 0
• P7: 1 + 0 = 0 + 1 = 1

1. T1 : Commutative Law
   • A + B = B + A
   • A B = B A
2. T2 : Associate Law
   • (A + B) + C = A + (B + C)
   • (A B) C = A (B C)
3. T3 : Distributive Law
   • A (B + C) = A B + A C
   • A + (B C) = (A + B) (A + C)
4. T4 : Identity Law
   • A + A = A
   • A A = A
5. T5 :
   • AB + AB' = A
   • (A+B)(A+B') = A
6. T6 : Redundance Law
   • A + A B = A
   • A (A + B) = A
7. T7 :
   • 0 + A = A
   • 0 A = 0
8. T8 :
   • 1 + A = 1
   • 1 A = A
9. T9 :
   • A' + A = 1
   • A'A = 0
10. T10 :
    • A + A'B = A + B
    • A (A' + B) = AB
11. T11 : De Morgan's Theorem
    • (A+B)' = A'B'
    • (AB)' = A' + B'

Examples
Prove T10: (a) A + A'B = A + B

1. Algebraically

   Code: Select all

   A + A'B = A1 + A'B               from T7(a)
           = A (1 + B) + A'B        from T8(a)
           = A + AB + A'B           from T3(a)
           = A + B (A + A')         from T3(a)
           = A + B                  from T8

2. Using the truth table

Using the laws given above, complicated expressions can be simplified.

thanks BRO....NEO.
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Thank you!