$F(A, B, C, D) = \Sigma m(0, 1, 2, 3, 8, 9, 10, 11)$
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
Consider the following four variable Boolean function in sum-of-product form
$F(b_3, b_2, b_1, b_0) = \sum(0, 2, 4, 8, 10, 11, 12)$.
where the value of the function is computed by considering $b_3b_2b_1b_0$ as a 4-bit binary number, where $b_3$ denotes the most significant bit and $b_0$ denotes the least significant bit. Note that there are no don't care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for F?
- Simplified Boolean equation for the following truth table is :
x y z F 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 The simplified form of a Boolean equation $(A\overline{B} + A\overline{B}C + AC)(\overline{A}\overline{C} + \overline{B})$ is :
Simplify the following Boolean expression.
E(E + F) + DE + D(E + F)
What is the value of \( \bar{F}\)?
\(F = AB + \bar{C}\bar{D} + \bar{B}D\)
