For the Boolean expression given belowF(A,B,C,D) = (2,3,6,7,9,11,13,15)d(A,B,C,D) = (0,4,5)(i) Plot the k-Map

1. Break Down the Problem

To plot the K-map for the given Boolean expression $F(A, B, C, D)$, we first identify the minterms and don't-care conditions.

Given:

• Minterms: $(2, 3, 6, 7, 9, 11, 13, 15)$
• Don't-care conditions: $(0, 4, 5)$

2. Relevant Concepts

A K-map for four variables (A, B, C, D) is represented in a 4x4 grid. The rows and columns are typically labeled as follows:

• Rows (AB): 00, 01, 11, 10
• Columns (CD): 00, 01, 11, 10

3. Analysis and Detail

We must fill in the K-map with 1s for the minterms and Xs for the don't-care conditions.

Step 1: Initialize the K-map

       CD
    00  01  11  10  
  -----------------
AB|  0   1   3   2  
00|  0   1   1   0  
01|  0   1   1   1  
11|  0   1   1   1  
10|  0   0   0   1  

Step 2: Place minterms and don't-cares

• Minterms: Place 1 in the locations corresponding to minterms.
• Don't-care: Place X in locations corresponding to don’t-care minterms.

K-map filled:

       CD
    00  01  11  10  
  -----------------
AB|  0   1   1   1  -> (rows correspond to minterms 2, 3, 6, 7)
00|  0   0   0   0  
01|  0   1   1   1  -> (rows correspond to minterms 9, 11, 13, 15)
11|  0   1   1   1  
10|  1   X   X   X  -> (0, 4, 5 are don't-care)

4. Verify and Summarize

We have placed the minterms as 1s and the don't cares as Xs in the correct positions. The K-map now helps visualize the combination of variables.

Final Answer

The K-map for the expression $F(A, B, C, D) = \Sigma(2, 3, 6, 7, 9, 11, 13, 15)$ with don't-care conditions $d(A,B,C,D) = \Sigma(0, 4, 5)$ is represented as follows:

       CD
    00  01  11  10  
  -----------------
AB|  0   1   1   1  
00|  0   0   0   0  
01|  0   1   1   1  
11|  0   1   1   1  
10|  1   X   X   X  

Where 1s represent the minterms and Xs represent don't-care conditions.