For three variables A, B, C, write the canonical Sum-of-Minterms for Y = Σ m (1, 3, 5, 7). Which expression is correct?
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AY = A'B'C + A'BC + AB'C + ABC
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BY = A'B'C + A'BC + ABC' + ABC
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CY = AB'C' + A'BC + AB'C + ABC
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DY = A'BC' + A'BC + AB'C + ABC
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EY = A'B'C' + A'BC + AB'C + ABC
Answer
Correct Answer: Y = A'B'C + A'BC + AB'C + ABC
Explanation
Introduction / Context:Canonical forms (Sum-of-Minterms and Product-of-Maxterms) are foundational in logic design, enabling systematic simplification and implementation using standard gates.
Given Data / Assumptions:
- Minterms: m(1, 3, 5, 7).
- Variables ordered as A (MSB), B, C (LSB).
Concept / Approach:
Each minterm index corresponds to the binary pattern of ABC. Index → bits: 1 → 001, 3 → 011, 5 → 101, 7 → 111. A 0 bit means complemented variable, a 1 bit means uncomplemented.
Step-by-Step Solution:
m1 (001) → A'B'Cm3 (011) → A'BCm5 (101) → AB'Cm7 (111) → ABCSum them: Y = A'B'C + A'BC + AB'C + ABCVerification / Alternative check:
Create a 3-variable truth table for indices 1,3,5,7 and confirm outputs are 1 only at those rows. K-map grouping also matches this canonical SOP.
Why Other Options Are Wrong:
- Options with any term having a wrong complemented/uncomplemented variable do not match the specified minterm index.
- Expressions including C' for indices whose LSB is 1 are invalid for this set.
Common Pitfalls:
Misordering variables (e.g., using C as MSB) and mixing up complements when translating index to literals.
Final Answer:
Y = A'B'C + A'BC + AB'C + ABC