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

We start by restating the problem: determine which expression is the proper simplification result of the given 4-input K-map, where cells labeled X are don't cares, and the options are as follows:
- Option 1: y = ad' + a'b'd + a'b'c
- Option 2: y = ad' + a'b'c
- Option 3: y = a'b' + ad'
- Option 4: y = a'b'd + ac'd' + b'cd'

Now we evaluate each option in turn, keeping in mind the goal of minimal sum-of-products form and that don’t-care cells can be used to form larger groups to reduce the expression.

Option 1: y = ad' + a'b'd + a'b'c
- The term a'b'c is present, which requires a'=1, b'=1, c=1. However, the combination a'b' already covers all minterms where a'=1 and b'=1, regardless of c or d, because a'b' is a broader term. Therefore a'b'c is redundant given a'b' is already included.
- Similarly, a'b'd is a subset of a'b' (since a'b' does not constrain d), so it is also redundant if a'b' were present. If the minimization aims for the smallest set of prime implicants, these narrower terms are not needed.
- Overall, this expression is not minimal because it contains redundant terms that are subsumed by a broader term (a'b').

Option 2: y = ad' + a'b'c
- This expression includes ad', which covers all minterms where a=1 and d'=1, but it lacks a term that covers the entire block where a'=1 and b'=1 (the a'b' region). The a'b'c term covers a'b' with c=1, but it does not cover the remaining a'b' combinations with c=0 or with d values that might be don’t cares or required to be covered.
- Without a'b' as a term, some required minterms (or don’t-care-turned-minterms) in the a'b' region may remain uncovered, leading to an incomplete cover. Hence this is not a correct minimal simplification.

Option 3: y = a'b' + ad'
- This expression combines a broad a'b' term, which covers all minterms with a'=1 and b'=1 regardless of c and d, with ad', which covers minterms with a=1 and d'=1 regardless of b and c. This pairing is a common minimal cover when the K-map allows grouping that yields these two implicants.
- Since a'b' already accounts for the a'=1, b'=1 region, and ad' covers the a=1, d'=1 region, there is no need for additional terms to cover those areas, and the result is typically minimal and correct if the don’t-care cells can be used to complete those groups.
- This option aligns with a clean, minimal sum-of-products form that commonly arises from proper K-map grouping with the given don’t cares.

Option 4: y = a'b'd + ac'd' + b'cd'
- The term a'b'd is a subset of a'b' (since a'b' would cover both d and d' states); including a'b'd does not add new coverage beyond a'b'. In minimization, such a redundant term is undesirable.
- The terms ac'd' and b'cd' introduce dependencies on c and d that do not appear in the compact two-term form suggested by the other options, and they do not reflect a minimal grouping that would typically emerge from a proper K-map simplification with don’t cares.
- Overall, this option seems to overcomplicate the expression and fails to present the most economical implicants.

Summary of reasoning across options:
- We look for a minimal set of implicants that cover all required 1s (and appropriately use Xs as don’t cares). If a broader implicant like a'b' is available, it subsumes narrower terms such as a'b'c or a'b'd, making those redundant. Likewise, any expression that omits essential broad implicants (like a'b' or ad') risks leaving some required minterms uncovered. Expressions that introduce extraneous, redundant, or non-minimal terms are not the proper simplified result.

In light of these analyses, Option 3 (y = a'b' + ad') provides the minimal, non-redundant cover consistent with typical K-map simplification using the given don’t-care cells.

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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