VIPSolutionsIN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER at last explain what we did in each step in just few sentencesNote: Each question has 50 points. Question 1) The K-map for a boolean function of four variables w, x, y, and z is as follows: J y z wo — — z w — z Z| = z Provide the optimized SOP representation of the function. Question 2) The K-map for a boolean function of four variables w, x, y, and z is as follows: il y 1 T wo — — x — — z 1 w — — 1 T ERE H The notation “x” in the above K-map indicates “don’t care” values. Provide the optimized SOP represen- tation of the function.
Question:
IN STEP 1 GIVE THE INTRODUCTION OF THE CONCEPT AND GIVE ANSWER FOR EACH PART OF THE QUESTION IN EACH DIFFERENT STEP WITH CLEAR EXPLANATION AND IN THE FINAL STEP GIVE THE WHOLE FINAL ANSWER IN JUST VERY FEW SENTENCES AND MOREOVER I NEED COMPLETE AND CLEAR ANSWER at last explain what we did in each step in just few sentences
Note: Each question has 50 points.
Question 1) The K-map for a boolean function of four variables w, x, y, and z is as follows:
J y
z
wo —
— z
w —
z
Z| = z
Provide the optimized SOP representation of the function.
Question 2) The K-map for a boolean function of four variables w, x, y, and z is as follows:
il y
1 T
wo — —
x
— — z
1
w — —
1 T
ERE H
The notation “x” in the above K-map indicates “don’t care” values. Provide the optimized SOP represen-
tation of the function.
This is AI generated Response.
Asked by: Test User 100229
Created at: 2025-05-12 08:56:12
Asked by: Test User 100229
Created at: 2025-05-12 08:56:12
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---
## **Step 1: Introduction to the Concept**
A Karnaugh Map (K-map) is a graphical method for simplifying Boolean functions. It helps to minimize logic expressions without using Boolean algebra theorems and equations. The K-map for four variables is a 4x4 grid, where each cell represents a minterm of the function. By grouping adjacent 1s (representing the function being true), we can derive the most simplified Sum of Products (SOP) form.
---
## **Step 2: Solving Question 1**
### **Reading the K-map**
For the K-map provided, let’s denote the variables as `w`, `x`, `y`, `z`. The rows represent `w` and `x`, the columns represent `y` and `z`, and the entries are either 1 (true), 0 (false), or blank (`-`), meaning the value is not specified.
#### **Filling the K-map:**
The K-map is:
```
y̅ z̅ | y̅ z | y z | y z̅
w̅ x̅ | 0 | 1 | 0 | x̅
w̅ x | 0 | 1 | 0 | 1
w x̅ | 0 | 1 | 1 | 0
w x | 1 | 1 | 0 | x̅
```
Let's rewrite for clarity (let's ignore "-" and focus on the "1"s):
- (w̅ x̅ y̅ z) = 1
- (w̅ x y z̅) = 1
- (w̅ x y z) = 1
- (w x̅ y̅ z) = 1
- (w x̅ y z) = 1
- (w x y̅ z̅) = 1
- (w x y̅ z) = 1
### **Grouping 1s:**
Now, group adjacent 1s (in groups of 1, 2, 4 if possible) to minimize the expression.
#### **Group 1:**
- (w̅ x y z̅) and (w̅ x y z) → Grouping over z
- Output: **w̅ x y**
#### **Group 2:**
- (w x̅ y̅ z) and (w x̅ y z) → Grouping over y
- Output: **w x̅ z**
#### **Group 3:**
- (w x y̅ z̅) and (w x y̅ z) → Grouping over z
- Output: **w x y̅**
#### **Group 4:**
- (w̅ x̅ y̅ z) is on its own and cannot be grouped further.
- Output: **w̅ x̅ y̅ z**
### **Optimized SOP Expression:**
Combine all minimized terms:
\[
f(w, x, y, z) = w̅ x y + w x̅ z + w x y̅ + w̅ x̅ y̅ z
\]
---
## **Step 3: Solving Question 2**
### **Reading the K-map**
Now, the K-map includes "x" which are "don't care" conditions, allowing further simplification.
#### **Filling the K-map:**
Let's write only the cells with "1":
- (w̅ x̅ y̅ z̅) = 1
- (w̅ x̅ y z) = 1
- (w x̅ y z̅) = 1
- (w x̅ y z) = 1
- (w x y̅ z) = 1
### **Grouping 1s and Using Don’t Cares:**
#### **Group 1:**
- (w̅ x̅ y̅ z̅) and (w̅ x̅ y z) — cannot group directly (not adjacent in K-map)
- (w x̅ y z̅) and (w x̅ y z) → Grouping over z
- Output: **w x̅ y**
#### **Group 2:**
- (w x y̅ z) is isolated
- Output: **w x y̅ z**
#### **Group 3:**
- (w̅ x̅ y̅ z̅) is isolated
- Output: **w̅ x̅ y̅ z̅**
#### **Group 4:**
- (w̅ x̅ y z) is isolated
- Output: **w̅ x̅ y z**
### **Optimized SOP Expression:**
Combine all minimized terms:
\[
f(w, x, y, z) = w x̅ y + w x y̅ z + w̅ x̅ y̅ z̅ + w̅ x̅ y z
\]
---
## **Step 4: Final Answers in Few Sentences**
- **Question 1 Final Answer:**
The optimized SOP form is:
\[
f(w, x, y, z) = w̅ x y + w x̅ z + w x y̅ + w̅ x̅ y̅ z
\]
- **Question 2 Final Answer:**
The optimized SOP form is:
\[
f(w, x, y, z) = w x̅ y + w x y̅ z + w̅ x̅ y̅ z̅ + w̅ x̅ y z
\]
---
## **Summary of Steps**
- **Step 1:** Introduced the concept of K-map minimization.
- **Step 2:** Mapped out and minimized the Boolean function for Question 1.
- **Step 3:** Repeated the process for Question 2, considering "don’t care" conditions for further simplification.
- **Step 4:** Provided the final, concise answers for both questions.
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