Overlapping groups

Also called: overlap in a K-map, K-map overlap, shared cells, overlapping K-map groups

K-map groups may share cells. Covering a 1 twice is harmless, since X + X = X, and overlapping often lets each group be larger.

On a K-map, two groups are allowed to overlap, sharing one or more cells. A 1 that is covered twice is still just a 1, because ORing a term with itself changes nothing: X + X = X, the idempotent law.

Overlap is often what makes a solution minimal. A 1 that touches two separate areas of 1s can join both groups, letting each be as big as possible. Without overlap, some 1 would be left in a smaller group of its own, giving a longer term.

The classic case is three 1s in an L shape. They can't be one group, but two overlapping pairs that share the corner cell cover them with 2-literal terms.

Overlap has one limit: every group must still earn its place. If all the 1s in a group are already covered by other groups, it is a redundant group and should be removed, even though overlapping it was legal.

Don't-cares can be shared between groups in the same way.

A\BC	00	01	11	10
0	1m0	1m1	0m3	0m2
1	0m4	1m5	0m7	0m6

Worked examples

Example

Sharing one cell

F = Σm(0, 1, 5), shown above.

1.
m1 touches both m0 (left) and m5 (below).
2.
Pair m0 m1 → . Pair m1 m5 → .
3.
F = . Without sharing m1, either m0 or m5 would need a 3-literal term.

Example

A pair overlapping a square

Simplify F = Σm(5, 7, 13, 14, 15).

AB\CD	00	01	11	10
00	0m0	0m1	0m3	0m2
01	0m4	1m5	1m7	0m6
11	0m12	1m13	1m15	1m14
10	0m8	0m9	0m11	0m10

1.
m5, m7, m13, m15 form a square → .
2.
m14 (1110) has one neighboring 1, m15. Pair them → .
3.
The two groups share m15. F = .

Common mistakes

Avoiding overlap and settling for a smaller group than the map allows.
Thinking a cell covered twice counts as 2. In Boolean OR, 1 + 1 = 1.
Overlapping so much that one group covers nothing new, which makes it redundant.

Practice Overlapping groups

Interactive questions with instant feedback and a worked solution for every wrong answer.

Boolean Simplification lesson full course

Karnaugh Maps lesson full course

And 1 more in the Karnaugh Maps lesson.

Practice exams

Learn it step by step

Overlapping groups is taught in Karnaugh Maps.

Karnaugh Maps →

All terms