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
Boolean Expression Conversion to Canonical POS Form
Understanding how to convert Boolean expressions between different forms like Sum of Products (SOP) and Product of Sums (POS) is fundamental in digital logic design. This solution will guide you through converting the given Boolean expression into its canonical Product of Sum (POS) form.
Boolean Expression Analysis
The given Boolean expression is:
\(F(x,y,z) = \bar xy\bar z + x\bar y\bar z + xy\bar z + xyz\)
This Boolean expression is presented in the Sum of Products (SOP) form, where each term is a min-term (a product of all variables in either complemented or uncomplemented form). To convert this into its canonical Product of Sums (POS) form, we first need to identify the decimal equivalents of the min-terms present in the given SOP expression.
| SOP Term | Binary Representation | Decimal Min-term |
|---|---|---|
| \(\bar xy\bar z\) | \(010_2\) | \(m_2\) |
| \(x\bar y\bar z\) | \(100_2\) | \(m_4\) |
| \(xy\bar z\) | \(110_2\) | \(m_6\) |
| \(xyz\) | \(111_2\) | \(m_7\) |
Thus, the given Boolean function \(F(x,y,z)\) can be represented in min-term notation as:
\(F(x,y,z) = \sum m(2, 4, 6, 7)\)
Identifying Max-terms for Canonical POS Form
The canonical Product of Sums (POS) form is a product of max-terms. Max-terms are essentially the complements of min-terms. For a Boolean function with three variables (\(x, y, z\)), there are \(2^3 = 8\) possible min-terms, ranging from \(m_0\) to \(m_7\).
The min-terms that are present in the given SOP form are \(m_2, m_4, m_6, m_7\). The min-terms that are not included in this set correspond to the max-terms that will form the canonical POS expression. These are the inputs for which the function's output is 0.
The missing min-terms are:
- \(m_0\) (Binary: 000)
- \(m_1\) (Binary: 001)
- \(m_3\) (Binary: 011)
- \(m_5\) (Binary: 101)
Therefore, the function \(F(x,y,z)\) can be expressed as a product of these corresponding max-terms:
\(F(x,y,z) = \prod M(0, 1, 3, 5)\)
Converting Max-terms to POS Products
To convert each max-term \(M_i\) into its sum term form, we follow a specific rule: for a given binary representation of \(i\), if a variable's bit is 0, the variable appears uncomplemented in the sum term. If the variable's bit is 1, the variable appears complemented in the sum term.
| Max-term | Binary Representation | POS Term (Sum Term) |
|---|---|---|
| \(M_0\) | \(000_2\) | \((x+y+z)\) |
| \(M_1\) | \(001_2\) | \((x+y+\bar z)\) |
| \(M_3\) | \(011_2\) | \((x+\bar y+\bar z)\) |
| \(M_5\) | \(101_2\) | \((\bar x+y+\bar z)\) |
By multiplying these individual sum terms together, we obtain the complete canonical Product of Sums (POS) form of the Boolean expression:
\(F(x,y,z) = (x+y+z)(x+y+\bar z)(x+\bar y+\bar z)(\bar x+y+\bar z)\)
Comparing with Options
Let's compare our meticulously derived canonical POS form with the provided options:
- Option 1: \(\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{y}} + {\rm{\bar z}}} \right)\)
- Option 2: \(\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\)
- Option 3: \(\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\bar y}} + {\rm{\bar z}}} \right)\)
- Option 4: \(\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{x}} + {\rm{y}} + {\rm{z}}} \right)\)
Our derived Boolean expression in canonical POS form perfectly matches Option 1.
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
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
Digital input signals A, B, C with A as the MSB and C as the LSB are used to realize the Boolean function F = m0 + m2 + m3 + m5 + m7, where mi denotes the ith minterm. In addition, F has a don’t care for m1. The simplified expression for F is given by:
