Given the following Karnaugh Map for a Boolean function $F(w, x, y, z)$:
yz wx 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
Which one or more of the following Boolean expression(s) represent(s) $F$?
The given Karnaugh Map (K-map) represents a Boolean function \(F(w, x, y, z)\). We need to identify which of the provided Boolean expressions correctly represents this function. Let us analyze the K-map step by step and derive the required Boolean expression:
| yz | 00 | 01 | 11 | 10 | |
| wx | 00 | 1 | 0 | 0 | 1 |
| 01 | 0 | 1 | 1 | 0 | |
| 11 | 0 | 1 | 1 | 0 | |
| 10 | 1 | 0 | 0 | 1 |
To derive the Boolean expression, we identify all cells in the K-map where the output is 1.
Reading the K-map:
- Cells \(w = 0, x = 0, y = 0, z = 0\): \(\overline{w}\overline{x}\overline{y}\overline{z}\)
- Cells \(w = 1, x = 0, y = 0, z = 0\): \(w\overline{x}\overline{y}\overline{z}\)
- Cells \(w = 0, x = 1, y = 0, z = 0\): \(\overline{w}x\overline{y}\overline{z}\)
- Cells \(w = 1, x = 0, y = 1, z = 1\): \(w\overline{x}yz\)
- Overlapping cells for \(y = 1, z = 0\) and \(y = 0, z = 1\) simplify to: \(x z\)
Thus, the Boolean expression covering all cells with 1 is:
\(\overline{w}\overline{x}\overline{y}\overline{z} + w\overline{x}\overline{y}\overline{z} + \overline{w}x\overline{y}\overline{z} + w\overline{x}yz + xz\)
Among the options provided, the following match this simplified Boolean expression:
- \(\overline{w}\overline{x}\overline{y}\overline{z} + w\overline{x}\overline{y}\overline{z} + \overline{w}x\overline{y}\overline{z} + w\overline{x}yz + xz\)
- \(\overline{x}\overline{z} + xz\) (this is a minimal expression derived by grouping terms with common literals)
Hence, the correct answers are the two expressions listed above.
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
