The given lists of minterms and don't care terms can be transformed to one shorter list of implicants using Boolean laws.
The Karnaugh-Veitch map:
From the map or by inspection of the given 14 of 16 possible terms, it is clear that there are only two terms where the expression is false: 1
and 2
.
Therefore, the expression can be written as
!(X0 & !X1 & !X2 & !X3) & !(!X0 & X1 & ! X2 & !X3)
Applying De Morgan's theorem:
(!X0 + X1 + X2 + X3) & (X0 + !X1 + X2 + X3)
Both factors share (X2 + X3)
which can thus be factored out:
(!X0 + X1) & (X0 + X1) + X2 + X3
This leads to the result:
!X0 & !X1 + X0 & X1 + X2 + X3
We implicitely set all don't care terms to true
at the beginning. This was a good guess, but cannot be analytically derived without try-and-error.
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