Adjacent minterms

Also called: adjacent, logically adjacent, logically adjacent minterms, adjacent maxterms, combining minterms, differ in one bit

Two minterms whose binary codes differ in exactly one bit. They combine into one product term with that variable removed.

Two minterms are adjacent when their row codes differ in exactly one bit, that is, their Hamming distance is 1. Then they agree on every variable but one, and that one is plain in one minterm and barred in the other. By combining, = X, the differing variable drops out.

In cube notation, the differing bit becomes a dash: m9 (1001) and m13 (1101) differ only in B, so they combine to 1-01, which is .

Compare bits, not numbers. Adjacency has nothing to do with being consecutive:

7 = 0111 and 8 = 1000 are consecutive but differ in all four bits.
3 = 011 and 5 = 101 differ by 2, a power of two, yet differ in two bits.
2 = 010 and 6 = 110 differ by 4 and are adjacent.

Combining can continue in rounds: two combined terms with their dashes in the same place, differing in one other bit, combine again. That is the heart of the quine mccluskey method and of Karnaugh maps, which arrange cells so adjacent minterms sit next to each other. The same idea works for adjacent maxterms in a POS.

Worked examples

Example

Combining a 4-variable pair

Combine m9 + m13 for inputs A, B, C, D.

1.
m9 = 1001 = and m13 = 1101 = .
2.
They differ only in B (bit 2 from the left).
3.
Replace that bit with a dash: 1-01.
4.
Result: , 3 literals.

Example

Two rounds

Combine m1, m3, m5 and m7 for inputs A, B, C.

1.
Codes: 001, 011, 101, 111.
2.
Round 1: m1 + m3 → 0-1, and m5 + m7 → 1-1.
3.
Round 2: 0-1 and 1-1 have their dash in the same place and differ only in A, so they combine to --1.
4.
Result: C. Four minterms, two variables removed.

Common mistakes

Treating consecutive row numbers as adjacent. 7 and 8 differ in every bit.
Treating a difference of a power of two as adjacency. 3 and 5 differ by 2 but in two bits.
Combining terms whose dashes are in different places, such as 0-1 and 01-.

Practice Adjacent minterms

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Boolean Simplification lesson full course

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Learn it step by step

Adjacent minterms is taught in Boolean Simplification and Karnaugh Maps.

Boolean Simplification →Karnaugh Maps →

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