Obtain a minimal SOP expression for f(A,B,C,D)=(sigma)m(2,4,5,6,7,9,11,12,13,14).
To obtain a minimal sum-of-products (SOP) expression for the given Boolean function f(A, B, C, D) = m(2, 4, 5, 6, 7, 9, 11, 12, 13, 14), you can follow these steps:
Step 1: Build a truth table
Create a truth table with columns for the inputs A, B, C, D, and the output function f. Assign 0s and 1s to each combination of inputs based on the given minterms.
| A | B | C | D | f |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 |
Step 2: Identify the minterms
From the truth table, identify the minterms that correspond to the rows with output f = 1. In this case, the minterms are: m(2, 4, 5, 6, 7, 9, 11, 12, 13, 14).
Step 3: Write the product terms
For each minterm, write a product term in the form (A * B * C * D). For example, m(2) corresponds to (A' * B' * C * D). It's important to use the complement (') on inputs that have a binary value of 0.
The product terms for the given minterms are:
(A' * B' * C' * D) - for m(2)
(A' * B * C' * D) - for m(4)
(A' * B * C * D) - for m(5)
(A' * B * C * D') - for m(6)
(A' * B * C' * D') - for m(7)
(A' * B' * C * D') - for m(9)
(A' * B' * C' * D') - for m(11)
(A * B' * C' * D) - for m(12)
(A * B' * C * D) - for m(13)
(A * B' * C' * D) - for m(14)
Step 4: Combine the product terms
Combine the product terms into a sum-of-products (SOP) expression by ORing them together. The resulting SOP expression is the minimal expression for the given function.
SOP expression:
(A' * B' * C' * D) + (A' * B * C' * D) + (A' * B * C * D) + (A' * B * C * D') + (A' * B * C' * D') + (A' * B' * C * D') + (A' * B' * C' * D') + (A * B' * C' * D) + (A * B' * C * D) + (A * B' * C' * D)
Simplified SOP expression:
A' * D + B' * C' * D + A' * B * C + A' * B * C' + B * C' * D' + B' * C * D' + C' * D * A + A * B' * C + B' * C * D
This is the minimal SOP expression for the given Boolean function f(A, B, C, D) = m(2, 4, 5, 6, 7, 9, 11, 12, 13, 14).