Understanding the Sum of Products (SOP) Expression

The problem provides a Boolean function F in the Sum of Products (SOP) form:

\(F = ABC + \bar A\bar B C + A\bar BC + \bar ABC + \bar A\bar B\bar C\)

We need to find the equivalent Product of Sums (POS) expression for this function.

Identifying Minterms from SOP

Each product term in the SOP expression represents a minterm. Assuming 3 variables (A, B, C), we can identify the minterms corresponding to the given expression:

• \(ABC\) corresponds to the binary combination 111, which is minterm \(m_7\).
• \(\bar A\bar B C\) corresponds to the binary combination 001, which is minterm \(m_1\).
• \(A\bar BC\) corresponds to the binary combination 101, which is minterm \(m_5\).
• \(\bar ABC\) corresponds to the binary combination 011, which is minterm \(m_3\).
• \(\bar A\bar B\bar C\) corresponds to the binary combination 000, which is minterm \(m_0\).

Therefore, the function F can be represented in canonical SOP form as:

\(F = \sum m(0, 1, 3, 5, 7)\)

Deriving Product of Sums (POS) from SOP

To find the equivalent POS expression, we first identify the minterms for which the function F is FALSE (i.e., 0). These are the minterms that are *not* included in the SOP expression.

The complete set of minterms for 3 variables (A, B, C) ranges from \(m_0\) to \(m_7\).

The minterms for which F is 0 are:

• \(m_2\) (010)
• \(m_4\) (100)
• \(m_6\) (110)

The function \(\bar F\) (complement of F) in SOP form is:

\(\bar F = \sum m(2, 4, 6)\)

Converting Missing Minterms to Maxterms

Each minterm corresponds to a specific maxterm in the dual function. The POS expression is the product of these maxterms.

We find the maxterms corresponding to the missing minterms (\(m_2, m_4, m_6\)):

• Maxterm \(M_2\) corresponds to \(\bar F = m_2 = 010\). The maxterm is formed by ORing the complements of the variables: \(A + \bar B + C\).
• Maxterm \(M_4\) corresponds to \(\bar F = m_4 = 100\). The maxterm is: \(\bar A + B + C\).
• Maxterm \(M_6\) corresponds to \(\bar F = m_6 = 110\). The maxterm is: \(\bar A + \bar B + C\).

Constructing the Equivalent POS Expression

The POS expression for F is the product of these maxterms:

\(F = M_2 \cdot M_4 \cdot M_6\)

\(F = \left( {A + \bar B + C} \right)\left( {\bar A + B + C} \right)\left( {\bar A + \bar B + C} \right)\)

Matching with Options

Comparing this result with the given options:

• Option 1: \(F = \left( {A + \bar B + C} \right)\left( {\bar A + B + C} \right)\left( {\bar A + \bar B + C} \right)\) - Matches our derived POS expression.
• Option 2: Contains terms like \(\bar C\) which are not in our derived maxterms.
• Option 3: Contains terms like \(\bar C\) which are not in our derived maxterms.
• Option 4: Contains terms like \(\bar C\) which are not in our derived maxterms.

Thus, the equivalent product of sums expression is given by Option 1.