GATE IT 2007 | Question: 78

Question

Consider the following expression

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

Which of the following Karnaugh Maps correctly represents the expression?

• A.
  
• B.
   
• C.
   
• D.
  

Comments

The question doesn't say that $X$ indicates a don't care. If $X$ indicates a don't care, then what do the remaining cells contain? $0$ or $1$ ??

Also, If $X$ indicates a don't care, then if option A is correct then so are $C,D.$

Correct Logic:

The given K-maps are a (not so common) way to representing a function on K-map by marking the minterm cells with $\checkmark$ or with $X.$

So, ALL the $X$ in the given K-maps actually represent minterms, Not don't cares.

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@Deepak Poonia Beautiful insight sir

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nice detailing

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Watch Solution here!   

Answer 1

$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. 

Comments

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

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@Arjun Sir, the question doesn't say that $X$ indicates a don't care. If $X$ indicates a don't care, then what do the remaining cells contain? $0$ or $1$ ??

Also, If $X$ indicates a don't care, then if option A is correct then so are $C,D.$

Correct Logic: The given K-maps are a (not so common) way to representing a function on K-map by marking the minterm cells with $\checkmark$ or with $X.$

So, ALL the $X$ in the given K-maps actually represent minterms, Not don't cares.

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https://youtu.be/coXIpBbLGOs?si=gKaBds3IC7KwmjMt

Answer 2

Answer is a.

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

Answer 3

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 😊

Comments

As Deepak sir mentioned above that , it is not written anywhere that X is Don't care hence consider it as minterm.

now if we expand our given SOP Expression adˉ+aˉcˉ+bcˉd

we will get minterms it as  m0,m1,m4,m5,m8,m10,m12,m13,m14

and we can see these minterms are only present in option A.