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Karnaugh map is a graphical method to reduce Boolean expressions. Most common type is four variables maps, like this one:

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Talking with coworkers we were wondering: If you swap two or more variables in the 4-variable K-map (keeping the same original expression), would it still give you the optimum expression? For example, if you swap A and C, obviously the map will change, but the reduction will be the same? Logic tell me the answer is yes, but I find it non-trivial to prove.

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Sometimes there are multiple 'optimal' solutions, and one map may suggest a different optimal solution than another. But like I said, in such cases you have multiple optimal solutions and, more to the point of your question, you can still generate the same formula in one map as you can in another, because as long as you know how you can group things, swapping variables does not make a difference.

And the proof for that is exactly that: as long as you specify which cells can be grouped together (and that becomes particularly important when you start getting more than 4 variables, because then groupings may occupy different spaces of the map, so toroidal considerations are not sufficient), then whatever grouping you can do in one map you can do in another.

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