Can you minimize a Boolean expression having don't care conditions using Boolean algebraic laws?

Question

I have to minimise a boolean expression in SOP form having don't care conditions. I can do this by k-map but can I do this only using Boolean algebraic laws.

Q. Sigma(0,2,3,5,6,7,8,9) + d(10,11,12,13,14,15)

Answer 1

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 trueat the beginning. This was a good guess, but cannot be analytically derived without try-and-error.