Which of these expressions is the proper simplification result of the 4-input K-map below? Cells with "X" are don't cares.

Question

BlackTom AI · Accepted Answer

Start by noting the given answer key indicates the simplification is y = a'b' + ab'.
Option 1 presents y = bc' + a'b'd + ac'd'. This form includes terms that involve c' and d' and do not align with the a and b dependencies seen in the target simplification. It cannot be reduced to a'b' + ab' without introducing new variables or missing essential coverage, so this is not equivalent to the stated result.
Option 2 presents y = a'b' + ab'. This expression captures exactly the combination of minterms where either a' and b' are true together, or a is true and b' is true. Algebraically, a'b' + ab' factors to b'(a' + a) = b', which matches the intended simplified form (and is consistent with the idea of don’t-cares covering the needed pairs). Therefore, this option is consistent with the given simplification.
Option 3 presents y = b'c' + a'b'd + ab'd'. Here, there is a b'c' term and two additional terms involving d. This introduces c' and d variables in ways that cannot be collapsed to a'b' + ab' without changing the function, so it does not match the target simplification.
Option 4 presents y = ab'c' + ab'd' + a'b'd. This mix includes ab' with c' and d' variants plus a'b'd, which again cannot be reduced to the simple two-term form a'b' + ab'. The presence of c' and d' in these terms prevents equivalence to the desired simplified expression.
In summary, only Option 2 yields a form that, upon standard Boolean algebra simplification, reduces to b' (which is equivalent to a'b' + ab'). The other options contain extra literals or terms that prevent such a reduction and thus do not match the requested simplification.

Which of these expressions is the proper simplification result of the 4-input K-map below? Cells with "X" are don't cares.

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