# Prime Implicants and Essential Prime Impliccants

12 votes
15.8k views

What are the prime implicants and essential prime implicants for the below questions ?

F(w, x, y, z) = ∑(1,2,5,7,12) + d(0,9,13)

Explain by drawing K-map.

Also explain the prime implicants and essential prime implicants with don't care condition .

retagged
2
essential prime inplicant-  (5,7) (0,2) (12,13) (1,5,9,13)

prime implicants- essential prime implicants + (0,1)  ???

## 2 Answers

23 votes

Best answer

K Map's with don't cares

A prime implicant is a rectangle of 1, 2, 4, 8, … 1’s or X’s not included in any one larger rectangle. Thus, from the point of view of finding prime implicants, X’s (don’t cares) are treated as 1’s.

An essential prime implicant is a prime implicant that covers at least one 1 not covered by any other prime implicant (as always). Don’t cares (X’s) do not make a prime implicant essential.

Here in the image prime implicants are marked with a rectangle.

Essential prime implicants have 1*.

This blue rectangle is not essential prime implicant because here there is no 1 which is only covered by single prime implicant. There is don't care(X) which is only covered by blue rectangle but in essential prime implicant, we only want a 1 which is not covered by another prime implicant.

Answer: 5 Prime implicants and 3 Essential prime implicants.

Good reads.
https://gateoverflow.in/80983/no-of-essential-prime-implicants
http://www-ee.ccny.cuny.edu/wwwn/yltian/Courses/EE210/EE210-Lecture7.pdf

selected
4
Yes, it is correct.

EPI is at position  8, 13 and 3 .

And there are only 3 EPI as we can see clearly from the diagram .

Total 5 Prime Implicants  are possible .
0
@Hemant Parihar, I have a doubt in the question which you posted as the first link.

In that if you see the K map of "vishwa ratna" which is the second answer, he has considered number 2 as essential PI. But the "1" at position "1001" can be covered by "X" above it and together it is a prime implicant. So how second one is essential prime implicant ?
2
Yes, I also think it should not be an essential prime implicant. 7 PI and 2 EPI. Let Habib confirm it. :)
0
@Hemant Parihar, According to your definition of prime implicant, if we consider all don't cares as 1's, then will a group of 4 don't cares (only don't cares) form a prime implicant? I don't think so.
1
@xylene it is not my definition, It is given the pdf link :) . Also I see few answer using this.

I also think that it should not be a prime implicants. Because we actually want to cover 1's. And we add don't care to it only when we are forming the big cube so that less literal in a term.

But I don't find any resource on it, If you find one do share :).
0
@Hemant Parihar, Can you please share the link where they have actually used don't cares like that ? I have not seen it anywhere till now. Even in the pdf link there is no example showing this.
1
Done @Hemant..:) ..
0

If the only one don't care is left in a group then it is considered as the PI or not ...See the last answer given by shruthi ,

at position 10 01 there is a don't care , so tell me it is EPI or PI

0 votes

prime implicant is the no. of all possible larger size subcube in k-map

essential prime implicant is the no. of larger size subcube which at least one cell without overlapping

here red one define E.P.I.

and red and yellow combined shows P.I.

1

@pawan, I have a doubt?

Are you sure this is essential prime implicant because here there is no 1 which is not covered by any other prime implicant? We don't care about X (don't care).

I agree that it will be preferred over the yellow marked prime implement because it has less literal.

0
I think (0,1) is not Prime emplicant.
0
yes becoz on my view to write minimal expression, essential prime implicant must present   in every possible minimal expression.and u know it will be prefer over yellow marked.

correct me someone if something wrong,,,,
1

@Pawan I don't think so it is prime implicant.
Let the question is this.

F(w, x, y, z) = ∑(1,2, 3, 5,7,12) + d(0,9,13)

only 3 is added in Sigma part.
Try it and say your answer.

2

essential prime implicant must present   in every possible minimal expression

No, EPI is not like that !

@pawan

An essential prime implicant is a prime implicant that covers at least one 1 which does not covered by any other prime implicant .

So at position 13, 8,3 have EPI .

see the best answer, it is correct .

0
thanks for pointing my mistake .....
1
@Bikram

Sir,

The best answer is correct, but what about the statement "every minimal expression should contain all essential prime implicants", isn't it true?
0

I think essential prime implicant must be present in minimal expression..

Why not? @Bikram

0
Why not ...
0

Sir in above comment you have told that

No, EPI is not like that !

0
essential prime implicant must present in every possible minimal expression No, EPI is not like that !

An essential prime implicant is a prime implicant that covers at least one 1 which does not covered by any other prime implicant .

So at position 13, 8,3 have EPI .

I don't think so it is prime implicant. see the best answer, it is correct .

Let the question is this. F(w, x, y, z) = ∑(1,2, 3, 5,7,12) + d(0,9,13) only 3 is added in Sigma part.

Try it and say your answer .

https://gateoverflow.in/80983/no-of-essential-prime-implicants
–1

$F=\Sigma(1,2,3,5,7,12)+d(0,9,13)$

3EPI and 4PI

0

Sir, i don't have any doubt in calculating PI and EPI and yes this answer is not correct and selected one is right..

But, I was asking that-

if there are some EPIs present then they all must be present in minimal expression along with some necessary prime implicant.

Because if we exclude any epi then the 'atleast one 1 which is not covered by any other prime implicant' is missed from our expression.

Correct?

0
0
K Map's with don't cares

A prime implicant is a rectangle of 1, 2, 4, 8, … 1’s or X’s not included in any one larger rectangle. Thus, from the point of view of finding prime implicants, X’s (don’t cares) are treated as 1’s.

An essential prime implicant is a prime implicant that covers at least one 1 not covered by any other prime implicant (as always). Don’t cares (X’s) do not make a prime implicant essential.

https://cse.sc.edu/~hoskinsw/classes/csce211/LecturesF15/Lecture8.pdf

https://gateoverflow.in/232643/epi-question?show=232643

http://media-lab.ccny.cuny.edu/wordpress/YLTCCNYHomepage/Courses/EE210/EE210-Lecture7.pdf

Here in the image prime implicants are marked with a rectangle. Essential prime implicants have 1*. This blue rectangle is not essential prime implicant because here there is no 1 which is only covered by single prime implicant. There is don't care(X) which is only covered by blue rectangle but in essential prime implicant, we only want a 1 which is not covered by another prime implicant.

5 Prime implicants and 3 Essential prime implicants.
0

What's wrong in this-

I don't found any error in this..

It is for $F=\Sigma(1,2,3,5,7,12)+d(0,9,13)$

Only $I,II,III$  are EPI and 4th group is considered as PI.

0

5 Prime implicants and 3 Essential prime implicants

Right for actual question..

But that (image in above comment) is other function).. it is just reply to your comment--

Let the question is this. F(w, x, y, z) = ∑(1,2, 3, 5,7,12) + d(0,9,13) only 3 is added in Sigma part.

Try it and say your answer .

0
F(w, x, y, z) = ∑(1,2,5,7,12) + d(0,9,13)

Explain by drawing K-map.

Also explain the prime implicants and essential prime implicants with don't care condition .

https://gateoverflow.in/?qa=blob&qa_blobid=5414455466690637946

Here in the image prime implicants are marked with a rectangle.

Essential prime implicants have 1*.

https://gateoverflow.in/?qa=blob&qa_blobid=5893376068232915243

This blue rectangle is not essential prime implicant because here there is no 1 which is only covered by single prime implicant.

There is don't care(X) which is only covered by blue rectangle but in essential prime implicant, we only want a 1 which is not covered by another prime implicant.

5 Prime implicants and 3 Essential prime implicants.....
Answer:

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