Simplified Boolean function f(A, B, C, D) = ∑(0, 2, 4, 5, 6, 7, 8, 10, 13, 15) in POS form is

Simplified Boolean Function in POS Form

Problem Statement: The problem requires simplifying the given Boolean function f(A, B, C, D), which is expressed in terms of minterms (Σm) as:

f(A, B, C, D) = Σ(0, 2, 4, 5, 6, 7, 8, 10, 13, 15)

The task is to express the given Boolean function in its Product of Sums (POS) form and determine the correct simplified expression among the given options. The correct answer is Option 4.

Steps to Solve:

To simplify the Boolean function in POS form, follow these steps:

Step 1: Identify the Minterms and Maxterms

The given function is expressed in minterms as Σ(0, 2, 4, 5, 6, 7, 8, 10, 13, 15). To convert this into POS form, we first determine the maxterms, which are the complements of the minterms. The maxterms are the indices not included in the minterm list:

Minterms: 0, 2, 4, 5, 6, 7, 8, 10, 13, 15

Maxterms: 1, 3, 9, 11, 12, 14

Step 2: Write the Maxterms in Standard POS Form

Each maxterm corresponds to the complement of the minterm. For a 4-variable function, the maxterms are expressed using the variables A, B, C, D in their complemented or uncomplemented forms:

• Minterm 1 → Maxterm: A + B' + C' + D'
• Minterm 3 → Maxterm: A + B' + C + D'
• Minterm 9 → Maxterm: A' + B + C' + D
• Minterm 11 → Maxterm: A + B' + C + D
• Minterm 12 → Maxterm: A' + B' + C + D'
• Minterm 14 → Maxterm: A + B' + C' + D

The POS form is the product (AND operation) of these maxterms.

Step 3: Simplify the POS Expression Using Boolean Algebra

Using Boolean algebra, the given function can be simplified step by step. After simplification, the Boolean function in POS form is:

f(A, B, C, D) = (B + D')(A' + B' + D)

Step 4: Match the Simplified Expression with the Given Options

The simplified POS expression matches Option 4. Hence, the correct answer is:

Option 4: (B + D')(A' + B' + D)