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How many implicants are present in the given k-map? enter image description here

I know that there are 6 prime implicant and 0 essential prime implicants.

My attempt:- by taking each 1 and X individually(as a product term) we get 6 implicants- size-1 implicant

by pairing we get 6 more implicants- enter image description here

so finally we have 12 implicants

Is it correct?

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(It has been a while since I took digital logic classes, but ...) Considering that the Xs are don't-cares they do not need to be included in the minimized logic expression and are there to be used only if needed. Also, prime implicants are supposed to maximize the number of cells (1s and X) they include, that being in order to reduce the expression. I would not count the groups 1 and 6, because they only link Xs, and the single cells such as groups 2 and 4 of your first picture, are not prime implicants because they don't maximize the number of cells they group. So, counting possible implicants (though not prime implicants) I guess you cound count the two lone 1s in the first picture and the groups 2, 3, 4 and 5 of the second picture posted.

K-map for the given problem

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  • \$\begingroup\$ Since 1 and 6 are prime implicants so shouldn't they be implicants? \$\endgroup\$
    – ashish
    Commented Jul 29, 2019 at 17:53
  • \$\begingroup\$ cseweb.ucsd.edu/classes/su15/cse140-a/slides/lec4_ann.pdf page 30 has a definition for implicants, through it, "A product term that has non-empty intersection with on-set F and does not intersect with off-set R". So, for a grouping to be an implicant, it has to have at least a 1, it cannot include the empty cells (zeros) and has to obey the k-map possible groupings \$\endgroup\$
    – jDAQ
    Commented Jul 29, 2019 at 18:03
  • \$\begingroup\$ Following that definition, the individual 1s are implicants, the individual Xs are not. The pairs of 1s and Xs are implicant if they obey the k-map possible groupings, but groups of Xs are not implicant if they don't include at least a 1. \$\endgroup\$
    – jDAQ
    Commented Jul 29, 2019 at 18:07
  • \$\begingroup\$ your edit adds a different question, and, it seems to be counting just single 1s as "implicants" and not all possible terms that are implicants, as your first question seems to define. Maybe it should be another question \$\endgroup\$
    – jDAQ
    Commented Jul 29, 2019 at 18:25
  • \$\begingroup\$ "A product term that has non-empty intersection with on-set F and does not intersect with off-set R". What does on-set and off-set means? \$\endgroup\$
    – ashish
    Commented Jul 29, 2019 at 18:34

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