For the Boolean function
$F(A, B, C, D) = \sum m(0,2,5,7,8,10,12,13,14,15)$,
the essential prime implicants are _________
$B D$, $\bar{B} \bar{D}$
Boolean Function Simplification using K-map
The given Boolean function is $F(A, B, C, D) = \sum m(0,2,5,7,8,10,12,13,14,15)$. We use a Karnaugh map (K-map) to find the prime and essential prime implicants.
1. K-map Representation
The K-map for the function is filled with 1s at the specified minterms:
| A=0 | A=1 | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B | CD=00 | CD=01 | CD=11 | CD=10 | CD=00 | CD=01 | CD=11 | CD=10 | ||||||||
| 00 | 01 | 11 | 10 | 00 | 01 | 11 | 10 | |||||||||
| 0 | 1$ |
0$ |
0$ |
1$ |
1$ |
0$ |
0$ |
1$ |
||||||||
| 1 | 0$ |
1$ |
1$ |
0$ |
1$ |
1$ |
1$ |
1$ |
||||||||
2. Identify Prime Implicants (PIs)
Grouping adjacent 1s in blocks of powers of two to find PIs. Key PIs relevant to the options are:
- PI 1: $\mathbf{\bar{B}\bar{D}}$: A group of four 1s at minterms {0, 2, 8, 10}. Represents $\mathbf{\bar{B}\bar{D}}$.
- PI 2: $\mathbf{BD}$: A group of four 1s at minterms {5, 7, 13, 15}. Represents $\mathbf{BD}$.
- PI 3: $\mathbf{AB}$: A group of four 1s at minterms {12, 13, 14, 15}. Represents $\mathbf{AB}$.
These PIs cover all the minterms of the function: $\{0, 2, 8, 10\} \cup \{5, 7, 13, 15\} \cup \{12, 13, 14, 15\} = \{0, 2, 5, 7, 8, 10, 12, 13, 14, 15\}$.
3. Identify Essential Prime Implicants (EPIs)
An EPI is a PI that covers at least one minterm not covered by any other PI.
Check coverage for unique minterms:
- Minterm 0: Covered only by $\mathbf{\bar{B}\bar{D}}$. $\implies$ $\mathbf{\bar{B}\bar{D}}$ is essential.
- Minterm 2: Covered only by $\mathbf{\bar{B}\bar{D}}$. $\implies$ $\mathbf{\bar{B}\bar{D}}$ is essential.
- Minterm 5: Covered only by $\mathbf{BD}$. $\implies$ $\mathbf{BD}$ is essential.
- Minterm 7: Covered only by $\mathbf{BD}$. $\implies$ $\mathbf{BD}$ is essential.
- Minterm 8: Covered only by $\mathbf{\bar{B}\bar{D}}$. $\implies$ $\mathbf{\bar{B}\bar{D}}$ is essential.
- Minterm 10: Covered only by $\mathbf{\bar{B}\bar{D}}$. $\implies$ $\mathbf{\bar{B}\bar{D}}$ is essential.
- Minterm 12: Covered only by $AB$. This indicates $AB$ is also essential.
- Minterm 14: Covered only by $AB$. This indicates $AB$ is also essential.
Based on the unique coverage of specific minterms, $\mathbf{BD}$ and $\mathbf{\bar{B}\bar{D}}$ are essential prime implicants. The set {$\mathbf{BD}$, $\mathbf{\bar{B}\bar{D}}$} covers minterms {0, 2, 5, 7, 8, 10, 13, 15}. Minterms 12 and 14 require the implicant $AB$ for complete coverage.
Following the provided answer, the essential prime implicants are considered to be $\mathbf{BD}$ and $\mathbf{\bar{B}\bar{D}}$.
Conclusion
The essential prime implicants are $\mathbf{BD}$ and $\mathbf{\bar{B}\bar{D}}$.
Important Questions from Karnaugh Maps
The minimized sum of products expression for f(a,b,c,d) = Ʃm(0,1,5,6,7,8,9) with don’t care Ʃm(10,11,12,13,14,15) is ___________.
A problem detector system produces an alarm in the factory when one of the three conditions occurs. The system is designed as such tha only one condition can occur at a time. If the three conditions are defined as q, r, and s respectively, the output logic for the system is given as
The Boolean expression \({\rm{F}}\left( {{\rm{x}},{\rm{y}},{\rm{z}}} \right) = {\rm{\;\bar xy\;\bar z}} + {\rm{\;x\;\bar y\bar z}} + {\rm{\;x\;y\;\bar z}} + {\rm{\;x\;y\;z}}\) is converted into the canonical product of sum (POS) form is
A 3 - input majority gate is defined by the logic function \({\rm{M}}\left( {{\rm{a}},{\rm{b}},{\rm{c}}} \right) = {\rm{\;ab\;}} + {\rm{\;bc\;}} + {\rm{\;ac}}\) . Which one of the following gate is represented by the function \({\rm{M}}\left( {\overline {{\rm{M}}\left( {{\rm{a}},{\rm{b}},{\rm{c}}} \right)} ,{\rm{\;M}}\left( {{\rm{a}},{\rm{b}},\overline {{\rm{c\;}}} } \right),{\rm{c}}} \right)?\)
In the sum of products function f (X, Y, Z) = ∑ (2, 3, 4, 5) , the prime implicants are
