Implicant: A group of one or more $1's$ which are adjacent and can be combined on a Karnaugh Map is called an implicant.
Prime Implicant: An implicant that is not a proper subset of any other implicant i.e. it is not completely covered by any single implicant.
Essential Prime Implicant: A prime implicant with at least one element that is not covered by one or more prime implicants.
Now, $f_{1}(w,x,y,z) = \Sigma_{m}(0,1,2,3,5,7 ,13 ,15)$
Now, we can count the number of implicants:
- The number of implicants of size $2^{0} = 8$
- The number of implicants of size $2^{1} = 10$
- The number of implicants of size $2^{2} = 3$
$\therefore$ The total number of implicants $ = 8 + 10 + 3 = 21.$
Prime implicants $ = 3,$ and essential prime implicants $ = 2.$
And, $f_{2}(a,b,c,d) = \Pi_{M}(0,1,2,4,5,8,10,13,14,15) =\Sigma_{m}(3,6,7,9,11,12)$
Now, we can count the number of implicants:
- The number of implicants of size $2^{0} = 6$
- The number of implicants of size $2^{1} = 4$
$\therefore$ The total number of implicants $ = 6 + 4 = 10.$
Prime implicants $ = 5,$ and essential prime implicants $ = 3.$
Now, the total number implicants $\alpha = 22 + 10 = 32,$ total number prime implicants $\beta = 3 + 5 = 8,$ and the total number of essential prime implicants $\gamma = 2 + 3 = 5.$
$\therefore$ The value of $\alpha - \beta + \gamma = 31- 8 + 5 = 28$
So, the correct answer is $28.$
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