Min-term and Max-term Don’t-Cares

A recent homework assignment in my digital electronics course must have made people ask my professor about how don’t-cares were represented in the functions, because he sent an email out explaining that \( + d \) represented don’t-cares for a min-term list and \( \cdot D \) represented don’t-cares for a max-term list. I would have assumed that to be the case anyway. However, it still struck up the question in my mind of why are the two represented differently anyway?

The reason I question this is because don’t-cares can be either 1’s or 0’s without altering the outcome of the function. So, in that case, whether we choose to express don’t-cares with a lower-case \( d \) or an upper-case \( D \), does it really matter? The don’t-cares will have the same values whether we’re looking at the function in terms of SOP or POS. I understand that it looks nicer to have a capitalized \( D \) with the capitalized \( M \) of the max-term list, but, in reality, the values of the don’t-cares remain the same within the same function whether we are looking at the min-term or max-term list of the function. Is that not entirely true? So why bother transitioning between the lower-case or upper-case to represent the numerical form in the function? It just seems pointless to me. A lower-case \( d \) isn’t going to throw the appearance of the expression off just because it is shown with a max-term list and ANDed as opposed to ORed with a min-term list.

\( f(A,B,C,D)=\prod M(1,2,3) \cdot d(0,4,5) \) works just as well as \( f(A,B,C,D)=\prod M(1,2,3) \cdot D(0,4,5) \)