Question
The simplification in minimal sum of product (SOP) of using K-map is:
Y = F(A, B, C, D) = ฮฃm(0,2,3,6,7) + ฮฃd(8,10,11,15)
Options:
- A. Y = AC + BD'
- B. Y = AC' + BD'
- C. Y = A'C' + B'D
- D. Y = A'C + B'D'
Answer: D
Explanation:
Using K-map grouping and including don't-care conditions:
- Group (2,3,6,7) → A'C
- Group (0 with 8 as don't-care) → B'D'
How do we get B'D'?
Given:
Y = ฮฃm(0,2,3,6,7) + ฮฃd(8,10,11,15)
Step 1: Look at minterm 0
Minterm 0 = 0000
A' = 1, B' = 1, C' = 1, D' = 1
So, minterm 0 is:
A'B'C'D'
Step 2: Use don't-care to expand grouping
Use don't-care term 8 = 1000
| Term | A | B | C | D |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 8 (d) | 1 | 0 | 0 | 0 |
Only A changes; B, C, and D remain constant.
So we eliminate A and keep common literals:
B'C'D'
Step 3: Further simplification
Using K-map grouping, we can form a larger group where:
- B = 0
- D = 0
- C varies
So C is eliminated.
Final simplified term:
B'D'
Summary
- Use don't-care terms to expand grouping
- Form the largest possible group in K-map
- Eliminate variables that change within the group
Final Result
B'D'