Identify a standard POS expression (variables A, B, C, D): Which option is in standard product-of-sums (POS) form, where each sum term includes all variables (possibly complemented)?
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A(A + B)(C + D)
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BAB + CD
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CA + BC
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D(A + B + C + D)(A + B' + C + D)
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E(A + C)(B + D')
Answer
Correct Answer: (A + B + C + D)(A + B' + C + D)
Explanation
Introduction / Context:In standard POS form for variables A, B, C, D, each sum (maxterm) must contain all four variables, either complemented or uncomplemented. This ensures a canonical structure that directly matches maxterm indices in a truth table.
Given Data / Assumptions:
- Variable set is A, B, C, D.
- Standard POS requires every sum term to list all variables.
Concept / Approach:Check each option: if any sum term omits one of the variables, it is not in standard POS (though it may be a valid non-standard POS). Only options where every sum term contains A, B, C, and D (with complements as needed) qualify as standard POS.
Step-by-Step Solution:
Step 1: Inspect option (A + B)(C + D): each term lacks variables → not standard.Step 2: AB + CD is SOP, not POS.Step 3: A + BC is neither standard POS nor SOP canonical.Step 4: (A + B + C + D)(A + B' + C + D) → each term has all variables → standard POS.Step 5: (A + C)(B + D') → missing variables per term → not standard.Verification / Alternative check:Expanding the correct option to maxterms demonstrates each term maps to a specific maxterm index set.
Why Other Options Are Wrong:
(A + B)(C + D): Non-standard POS (terms incomplete).AB + CD: SOP, not POS.A + BC: Mixed form, not standard POS.(A + C)(B + D'): Non-standard POS.Common Pitfalls:Confusing valid POS with standard POS; “standard” implies all variables present per sum term.
Final Answer:(A + B + C + D)(A + B' + C + D)