Q.Using the four-variable K-map shown below (which contains don't-care conditions marked X), simplify the expression first without using the don't-cares and then with the don't-cares, and compare the two results.

Treating the X cells as 0 forces two small pairs, but letting the X cells act as 1s lets the same 1s join into two larger quads, dropping one literal from each term. …

[!TLDR]

A don't-care X can be read as 1 or 0 to enlarge groups; using them here shrinks $\overline{A}BD + \overline{A}C\overline{D}$ to the simpler $BD + C\overline{D}$.

The map

Don't-care conditions

A don't-care is an input combination that cannot occur, marked X. When grouping, an X may be treated as a 1 or a 0 — whichever helps make a larger group.

Without the don't-cares (every X counts as 0) only the real 1s can be grouped, giving two pairs:

• Row $\overline{A}B$, columns $\overline{C}D$ and $CD$: $C$ changes; constants $A=0,\ B=1,\ D=1$ give $\overline{A}BD$.
• Column $C\overline{D}$, rows $\overline{A}\,\overline{B}$ and $\overline{A}B$: $B$ changes; constants $A=0,\ C=1,\ D=0$ give $\overline{A}C\overline{D}$.

With the don't-cares (chosen Xs count as 1) the same 1s extend into two quads: …