A 4-to-1 multiplexer selects one of four data inputs (D0-D3) to pass through to the output, based on two select lines (S1, S0). The standard textbook equation for it is usually just stated as correct without ever checking whether it can actually be simplified further. Treating all six signals as independent K-map variables and checking directly: it can't.
The standard equation
Each data input passes through when the select lines match its corresponding binary code:
Four terms, one per data input, each gated by the select-line combination that should route that input to the output.
Does this actually simplify further?
Treating S1, S0, D0, D1, D2, D3 as six independent Boolean variables — exactly what this solver's 6-variable mode handles — the standard equation maps to 32 minterms. Run through the full minimization: the result comes back as exactly the same four terms, just reordered. There's no smaller expression for this function; the textbook form isn't a convenient approximation, it's the genuine K-map-minimal answer, verified rather than assumed.
Why it doesn't simplify
Each of the four terms depends on a distinct, non-overlapping combination of the select lines — S1′S0′, S1′S0, S1S0′, S1S0 are the four possible select-line states, and no two of them can ever be true at once. With no overlap between the required regions and no shared structure to factor out, there's no adjacency for a K-map to exploit — every term really is doing necessary, distinct work.
The practical design takeaway
Not every "obvious" digital design equation is actually minimal until it's checked — but multiplexer logic, built directly from mutually exclusive select-line combinations, is one of the cases where the straightforward construction already is the minimal one. Worth confirming rather than assuming, especially on wider multiplexers (8-to-1, 16-to-1) where more terms means more opportunity for an actual simplification to exist.
Verify a multiplexer equation of your own directly on the 6-variable solver.