2-Variable Karnaugh Maps

Two variables means four minterms, so the smallest Karnaugh map is a 2x2 grid: one bit assigned to the rows, one to the columns. There's no room yet for a multi-digit Gray code sequence (each axis is simply 0 then 1), but the map already has the shape every larger map will share, including the split corner cell in the top-left that holds the row and column variable names.

Worked Example: F = A + B

Below is the map for F(A, B) = A OR B. Three of the four cells hold a 1. Click a 1, click another adjacent 1, then press Check group once you think you've found a legal pair. A legal group has to be a rectangular block sized 1, 2, or 4 cells here, since 2x2 is as big as this map gets.

Reading the Groups

Two groups of size 2 cover every 1 in the map: the pair where B stays 1 while A changes (giving the term B), and the pair where A stays 1 while B changes (giving the term A).

The cell where A = 1 and B = 1 belongs to both groups. Overlap is completely legal here, the goal is to cover every 1 with the fewest, largest groups, not to partition the map into disjoint pieces.

Why This Is the Same Algebra as Before

Each group of 2 cells is doing exactly what the identity from the previous tutorial does: AB' + AB = A(B' + B) = A. Grouping visually and factoring algebraically are the same operation, the grid just makes it obvious which minterms happen to differ in one variable.

A'B + AB = B(A' + A) = B
AB' + AB = A(B' + B) = A

OR the surviving terms together and you get F = A + B back out, the exact function you started with, just derived visually instead of algebraically. For a function this small that's not a huge win by itself, but the same procedure keeps working as the variable count climbs, which is the entire point of learning it here first.

A 2x2 grid is too small to ever need to wrap around an edge, every cell already touches every other cell along at least one side. That changes at three variables.