$F(A, B, C, D) = \Sigma m(0, 1, 2, 3, 8, 9, 10, 11)$
Consider $A$ as MSB, $D$ as LSB. Which one of the following options represents the minimal sum of products form for the above function?
Note: $+$ is OR operation, $.$ is AND operation, $'$ is NOT operation
Understanding the Boolean Function and Minterms
The given Boolean function is defined by its minterms: $F(A, B, C, D) = \Sigma m(0, 1, 2, 3, 8, 9, 10, 11)$. This signifies that the function output is 1 for these specific input combinations. Variable $A$ is the MSB and $D$ is the LSB.
Constructing the Karnaugh Map (K-map)
A 4-variable Karnaugh map (K-map) is constructed to simplify the function. The map uses AB for rows and CD for columns. Cells corresponding to the given minterms are marked with '1'.
| CD ↓ &brbar AB → | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| ($A'B'$) | ($A'B$) | ($AB$) | ($AB'$) | |
| 00 ($C'D'$) | 1 | 0 | 0 | 1 |
| 01 ($C'D$) | 1 | 0 | 0 | 1 |
| 11 ($CD$) | 1 | 0 | 0 | 1 |
| 10 ($CD'$) | 1 | 0 | 0 | 1 |
Grouping '1's for Minimal SOP
To achieve the minimal Sum of Products (SOP) form, adjacent '1's in the K-map are grouped into the largest possible blocks (powers of 2).
- Group 1: The first column (cells corresponding to $A'B'$) contains four '1's (minterms 0, 1, 2, 3). This yields the term $A'B'$.
- Group 2: The last column (cells corresponding to $AB'$) contains four '1's (minterms 8, 9, 10, 11). This yields the term $AB'$.
These two groups are essential and collectively cover all '1's.
Deriving the Minimal Sum of Products (SOP)
The minimal SOP is the OR combination of the terms derived from the groups.
Function $F = A'B' + AB'$
Simplifying the Expression
Boolean algebra rules are applied for simplification.
Factor out the common term $B'$:
$F = B'(A' + A)$
Using the identity $X + X' = 1$, we get $A' + A = 1$.
$F = B'(1)$
Using the identity $X \cdot 1 = X$, the simplified expression is:
$F = B'$
This minimal SOP form matches Option B.
Important Questions from Minimization of Boolean Expression
What is the value of \( \bar{F}\)?
\(F = AB + \bar{C}\bar{D} + \bar{B}D\)
Simplify the following Boolean expression.
E(E + F) + DE + D(E + F)
Which statement(s) is/are correct regarding the Boolean algebra?
I. It facilitate the analysis and design of digital circuits.
II. Expresses in algebraic form the input-output relationship of logic diagram.
The input-output relationship of the binary variable for each gate can be represented in tabular form by a _______.
What is the simplified expression for the Boolean function F(A, B, C, D) = Σ(0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14) using the K - map method?
