Boolean Algebra · step 10 of 13

Don’t-care conditions in minimisation

A don’t-care is an input combination that cannot occur, or whose output nobody cares about — BCD codes 1010 through 1111 are the standard example. Marked X on a K-map, each may be treated as 1 or 0 at your convenience, whichever makes groups larger. Used well they cut gate count substantially; used carelessly they add terms that cover nothing useful.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Mark don’t-cares as XEnter them on the K-map alongside the 1s and 0s, from the stated impossible or irrelevant combinations.
Include Xs only to enlarge a groupTreat an X as 1 when it lets a group of 2 become a group of 4. Otherwise treat it as 0.
Never form a group of Xs aloneA group covering no genuine 1 adds a term for nothing.
Cover all 1s, not all XsThe requirement is that every 1 is covered. Xs left uncovered are entirely fine.

Worked example

Minimise f(x,y,z) = Σm(1,3,7) with don’t-cares d(0,2).

Minterms 1 and 3 have x = 0, z = 1; adding don’t-cares 0 and 2 fills the whole x = 0 half.
That gives a group of four covering x = 0, yielding the term x′.
Minterm 7 (x=1,y=1,z=1) pairs with minterm 3 (x=0,y=1,z=1), giving yz.
All three 1s are covered; the unused don’t-cares are ignored.

Answer. f = x′ + yz — the don’t-cares turned a two-cell group into a four-cell one and removed a literal.

Where marks get dropped

These are the specific errors that cost credit on don’t-care conditions in minimisation questions — QED's rubric penalises each of them separately.

Treating a don’t-care as a 1 that must be covered. Its coverage is optional, and covering it "just in case" can cost a whole product term.
Making a group entirely of Xs, which contributes a term covering no required output.
Assuming the don’t-care values are free at runtime. If the combination can actually occur, the output really is unspecified — and that may be a hardware bug.

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Don’t-care conditions in minimisation — frequently asked questions

Where do don’t-cares come from?

Impossible input combinations (BCD digits above 9), inputs guaranteed by an earlier stage, or outputs that are ignored downstream because another signal gates them.

Do don’t-cares always help?

They never hurt, since treating them as 0 recovers the original problem. They help whenever including one enlarges a group.

How are they written in Σ notation?

f = Σm(1,3,7) + d(0,2), listing the required 1s and the don’t-cares separately.

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

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