Write five theorem of Boolean Algebra and simplify F = (A+B)' (A+B')'

Q.) Write five theorem of Boolean Algebra and simplify F = (A+B)' (A+B')'

 Subject: digital circuit and design

Boolean Algebra is a branch of algebra that deals with variables that have two possible values: true (1) and false (0). It is used extensively in digital electronics and computer science. Here are five fundamental theorems of Boolean Algebra:

  1. Identity Law

    • For AND: ( A \cdot 1 = A )
    • For OR: ( A + 0 = A )

  2. Null Law

    • For AND: ( A \cdot 0 = 0 )
    • For OR: ( A + 1 = 1 )

  3. Idempotent Law

    • For AND: ( A \cdot A = A )
    • For OR: ( A + A = A )

  4. Inverse Law

    • For AND: ( A \cdot A' = 0 )
    • For OR: ( A + A' = 1 )

  5. Complement Law

    • For AND/OR: ( (A')' = A )

Now, let's simplify the given Boolean expression using these theorems:

( F = (A+B)' (A+B')' )

Step 1: Apply De Morgan's Theorem to the complements De Morgan's Theorem states that:

  • ( (A \cdot B)' = A' + B' )
  • ( (A + B)' = A' \cdot B' )

Using this theorem, we can rewrite the expression as: ( F = (A' \cdot B') (A' + B) )

Step 2: Apply the Distribution Law The Distribution Law allows us to expand the expression as follows:

  • ( A \cdot (B + C) = A \cdot B + A \cdot C )
  • ( A + (B \cdot C) = (A + B) \cdot (A + C) )

Applying this to our expression, we get: ( F = A' \cdot B' \cdot A' + A' \cdot B' \cdot B )

Step 3: Apply the Idempotent Law According to the Idempotent Law, ( A \cdot A = A ) and ( A + A = A ). Thus, we can simplify ( A' \cdot A' ) to ( A' ): ( F = A' \cdot B' + A' \cdot B' \cdot B )

Step 4: Apply the Null Law and the Identity Law The Null Law states that ( A \cdot 0 = 0 ) and the Identity Law states that ( A + 0 = A ). Since ( B' \cdot B = 0 ), we can simplify the expression further: ( F = A' \cdot B' + 0 ) ( F = A' \cdot B' )

Step 5: Final Simplification After applying the above laws, we have the final simplified expression: ( F = A' \cdot B' )

Here's a summary table of the steps and theorems applied:

Step Expression Theorem/Law Applied
1 ( (A+B)' (A+B')' ) De Morgan's Theorem
2 ( (A' \cdot B') (A' + B) ) Distribution Law
3 ( A' \cdot B' \cdot A' + A' \cdot B' \cdot B ) Idempotent Law
4 ( A' \cdot B' + A' \cdot B' \cdot B ) Null Law and Identity Law
5 ( A' \cdot B' ) Final Simplification

The final simplified expression for ( F ) is ( A' \cdot B' ), which is the product of the complements of ( A ) and ( B ).