The definition says that we can use a don't care as either 1 or 0. It seems that if we consider a don't care as 1, we can't consider it as 0 while computing product of sum from the same k-map. Is it true? Any explanation on this stating the actual significance of don't care would be highly appreciated.
Don't care means what it says, we don't care what the output is under that combination of inputs.
So if we treat it as a 1 when doing SOP and a 0 when doing POS we will end up with two results which are not equivilent to each other but which are both valid soloutions to the original problem described by the k-map.