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30 votes
30 votes

Consider the following expression

$a\bar d + \bar a\bar c + b\bar cd$

Which of the following Karnaugh Maps correctly represents the expression?



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in Digital Logic edited by
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3 Answers

19 votes
19 votes
Best answer

$a\bar d + \bar a \bar c+b \bar c d = \overset{m_8}{a\bar b \bar c \bar d} + \overset{m_{10}}{a\bar b  c \bar d} +\overset{m_{12}}{a b \bar c \bar d} + \overset{m_{14}}{a b  c \bar d} $

$\qquad + \overset{m_0}{\bar a \bar b \bar c \bar d} + \overset{m_4}{\bar a  b \bar c \bar d}+ \overset{m_1}{\bar a \bar b \bar c  d}+ \overset{m_5}{\bar a  b \bar c  d}$

$\qquad + \overset{m_5}{\bar ab \bar c d}+ \overset{m_{13}}{ab\bar c d}$

When we minimize a K-map, we can assume either $0$ or $1$ for don't cares. But here they have asked for the expression represented by the K-map. So we can consider $X$ as $1$ and not as a don't care. Also the given expression is equivalent to the above K-map but not the minimal one. Minimal expression will be $\bar a \bar c + b\bar c+ a\bar d.$

Hence, Option A. 

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3 Comments

isn’t the K-map is wrong mapped because we can make square in 2 and 3 rd row with four X. please explain @Deepak Poonia sir

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@chinmay_rajpurohit

You are correct but read the question carefully. There is one expression with 3 literals, so we have to make a square with only 2 required don't care literals. In the question, it is not asking for the optimized K-Map. It asks which K-Map can be represented by the given boolean function. You have to check the K-Maps, which fulfill all the expressions by selecting desired don't care literals.

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Yeah. I found out that later. Thanks for answering sir
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11 votes
11 votes

Answer is a.

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4 Comments

But we can still derive this expression using (c) and (d) as well right? They didn't ask for a minimal don't care set of K-map or something.

Please explain how are you being confident of the option (a)?
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Even the given expression is not the minimal for option A.

d can be removed from bc'd.
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@Sunit Acharya 

I think the expression should be able to cover every minterm . And it does not cover every minterm from the kmap c and d . 

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here X represents 1 or don't care
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1 vote
1 vote
If you add all the missing literal in the expression then this expression will give the 9 minterms or you can say

Sum(m0,m1,m4,m5,m10, m12,m13,m14, m8)

 

Now option C and option D have eliminated becoz it has 10 and 11 minterms respectively

 

Now it's a game between option A and option B . Now look at 0101 or m5 in option A which is minterm of given expression but it is not present in option B .

 

Warning : X (don't care ) can be treated as 0 or 1 as per our requirement .

So final answer is A.

Hope u like it 😊
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