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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 .
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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

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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.

Answer:

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