State thevenin's theorem of Boolean algebra and simplify F = (A + B)' (A + B')'

Q.) State thevenin's theorem of Boolean algebra and simplify F = (A + B)' (A + B')'

 Subject: digital circuit and design

Thevenin's Theorem of Boolean Algebra:

Thevenin's theorem states that any Boolean function can be represented by a series of two-input AND gates followed by a single inverter. This representation is unique and can be obtained by applying the following steps:

  1. Express the Boolean function in sum-of-products (SOP) form.
  2. Group the terms in the SOP form such that each term contains a common variable.
  3. Draw a two-input AND gate for each group of terms.
  4. Connect the outputs of the AND gates to a single inverter.

The output of the inverter is the Thevenin equivalent of the Boolean function.

Simplification of F = (A + B)' (A + B')':

Using Thevenin's theorem, we can simplify the Boolean function F = (A + B)' (A + B')' as follows:

  1. Express F in SOP form:
F = (A + B)' (A + B')'
= (A' + B') (A' + B)
= A'A' + A'B + B'A' + B'B
= A' + B'
  1. Group the terms in the SOP form:
F = (A' + B')
  1. Draw a two-input AND gate for the group of terms:
       A'     B'
        \    /
         AND
          |
          F
  1. Connect the output of the AND gate to a single inverter:
       A'     B'
        \    /
         AND
          |
         INV
          |
          F

The output of the inverter is the Thevenin equivalent of F.

Conclusion:

Using Thevenin's theorem, we have simplified the Boolean function F = (A + B)' (A + B')' to its Thevenin equivalent, which is a single inverter connected to a two-input AND gate. This representation is useful for analyzing and designing Boolean circuits.