BOOLEAN ALGEBRA

BOOLEAN ALGEBRA

1. NOT GATE (Inverter)

   • A NOT gate, also known as an inverter, is a fundamental logic gate that inverts its input. If the input is 1 (true), the output is 0 (false), and if the input is 0 (false), the output is 1 (true)

2. AND GATE

   • The AND gate produces a "1" (high) output only if all its inputs are "1" (high). Otherwise, it outputs a "0" (low)

3. OR GATE

   • An OR gate is a digital logic gate that outputs "true" (or 1) if any of its inputs are "true" (or 1), and "false" (or 0) only if all inputs are "false" (or 0)

4. NAND GATE

   • A NAND gate is a logic gate that outputs a LOW (0) if and only if all its inputs are HIGH (1). Otherwise, it outputs HIGH (1)

5. NOR GATE

   • A NOR gate is a digital logic gate that outputs 1 only when all its inputs are 0. Otherwise, it outputs 0

6. XOR GATE

   • A Logic XOR (Exclusive OR) gate is a digital logic gate that outputs a true (1) value if the two inputs are different, and a false (0) value if they are the same

7. Definition

   • Boolean algebra is a branch of algebra dealing with logical values, specifically true and false, often represented as 1 and 0

8. Truth Table

   • Definition
     • A truth table in logic gates is a tabular representation that shows the output of a logic gate for all possible combinations of its input values. It's a way to define the function of a logic gate by listing all possible input and output states

9. De Morgan's Law

   • Definition
     • De Morgan's Law in Boolean algebra states that the complement of a product is the sum of the complements, and the complement of a sum is the product of the complements
   • Theorem
     • First Theorem
       • (A + B)' = A' . B'
     • Second Theorem
       • (A.B)' = A' + B'

10. Boolean Theorem

    • Duality Theorem
      • Theorem states that the dual of a Boolean function can be obtained by swapping AND and OR operators and replacing 0 with 1 and 1 with 0
    • Commutative Law
      • A + B = B + A
      • A . B = B . A
    • Associative Law
      • (A + B) + C = A + (B + C)
      • (A . B) . C = A . (B . C)
    • Distributive Law
      • A . (B + C) = (A . B) + (A . C)
      • A + (B . C) = (A + B) . (A + C)
    • Identity Law
      • A . 0 = 0
      • A . 1 = A
      • A + 0 = A
      • A + 1 = 1
    • Complement Law
      • A . A' = 0
      • A + A' = 1
    • Absorption Law
      • A + (A . B) = A
      • A . (A + B) = A