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I am having some trouble understand essential prime implicants. So if a minterm is not covered by another overlapping rectangle, then that is an EPI. However, if we make a K-map for $f(x,y,z)=xy+xz'+y'z$, we have minterms m4, m6, and m7 not covered by overlapping rectangles. If what I did is correct, then there should be a total of only 2 rectangles in the table-one horizontal one that completely encapsulated the 2nd row, and a vertical rectangle for the 2nd column. And so it is a rule that EPI must appear in the minimal sum of products, in which case I'd have a lot of terms, but in fact I only see that it simplifies to $y'z+x$. What am I missing here?

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Prime implicant of $f$ is an implicant that is minimal - that is, if the removal of any literal from product term results in a non-implicant for $f$ .

Essential prime implicant is an prime implicant that cover an output of the function that no combination of other prime implicants is able to cover.

Let's observe these two pictures bellow.The first picture represents all prime implicants while the second picture represents only essential prime implicants:

enter image description here enter image description here

In your specific case (picture bellow) both of the prime implicants, $x$ and $y'z$ , are essential prime implicants also.

enter image description here

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Diagrams! You have so much power. –  Ziggy Nov 11 '12 at 21:04
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