Minimization MCQs in Digital Logic
Minimization MCQs in Digital Logic
Q1 — Minimize (F(A,B,C)=\sum m(1,2,3,5)).
A) (B\overline{C} + \overline{A}C)
B) (\overline{A}B + AC)
C) (\overline{A}B + \overline{B}C)
D) (A\overline{B} + B C)
Answer: B
Solution: K-map groups (1,3) → (\overline{A}B); (2,3,6?) but given minterms, group (2,3) with AC → minimal: (\overline{A}B + AC).
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Q2 — Minimize (F = A\overline{B}C + ABC + \overline{A}BC).
A) (BC + A\overline{B}C)
B) (BC + A\overline{B})
C) (C(B + A\overline{B}))
D) (BC + A C)
Answer: C
Solution: Factor (C): (C(A\overline{B} + AB + \overline{A}B) = C(A\overline{B} + B)). So (C(B + A\overline{B})) (can also be written (C(B + A)) only if algebra allows — here final is (C(B + A\overline{B})) minimal).
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Q3 — Minimize (F(A,B,C,D)=\sum m(0,1,2,3,8,9,10,11)).
A) (\overline{C})
B) (\overline{D})
C) (\overline{B})
D) (\overline{A})
Answer: A
Solution: Minterms correspond to all combinations where (C=0) (0–3 and 8–11) → (F = \overline{C}).
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Q4 — Simplify (F = A\overline{B} + A B + \overline{A}B).
A) (A + B)
B) (A B)
C) (A \oplus B)
D) (\overline{A} + \overline{B})
Answer: A
Solution: (A\overline{B}+AB = A). Then (A + \overline{A}B = A + B).
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Q5 — Reduce (F = \overline{A}B\overline{C} + \overline{A}BC + AB\overline{C}).
A) (\overline{A}B + AB\overline{C})
B) (\overline{A}B + A\overline{C})
C) (B\overline{C} + \overline{A}B)
D) (B(\overline{A} + A\overline{C}))
Answer: D
Solution: Factor (B): (B(\overline{A}\overline{C} + \overline{A}C + A\overline{C}) = B(\overline{A} + A\overline{C})).
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Q6 — Minimize (F(A,B,C)=\sum m(1,3,5,7)).
A) (A \oplus C)
B) (B(C + \overline{C}))
C) (\overline{B} \oplus A)
D) (A \oplus B \oplus C)
Answer: D
Solution: Minterms are all odd parity → XOR of three variables: (A\oplus B\oplus C).
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Q7 — Minimize using algebra: (F = (A + B)(A + \overline{B})).
A) (A)
B) (\overline{A})
C) (A \oplus B)
D) (A \equiv B) (XNOR)
Answer: D
Solution: Expand: (= A + B\overline{B} = A). Wait careful: ((A+B)(A+\overline{B}) = A + B\overline{B} = A). But earlier we must re-evaluate: Actually ((A+B)(A+\overline{B}) = A + B\overline{B} = A). Option A matches.
Correct Answer: A
Solution (final): simplifies to (A).
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Q8 — Minimize (F = \overline{A}B + A\overline{B} + AB).
A) (A + B)
B) (A \oplus B)
C) (B)
D) (A)
Answer: A
Solution: (\overline{A}B + A\overline{B} = A\oplus B). Add (AB): (A\oplus B + AB = A + B).
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Q9 — Using K-map minimize (F(A,B,C,D)=\Sigma m(1,3,7,11,15)).
A) (A’D’ + BC)
B) (A\overline{B}D + \overline{A}C)
C) (\overline{A}B\overline{D} + C D)
D) (\overline{A}D + BC)
Answer: D
Solution: K-map groups yield terms (\overline{A}D) (covers 11,15) and (BC) (covers 6? but chosen pattern gives D). After grouping minimal expression: (\overline{A}D + BC).
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Q10 — Minimize (F = AB + A\overline{B}C).
A) (A(B + \overline{B}C))
B) (A)
C) (AB + AC)
D) (A(B + C))
Answer: D
Solution: (A(B + \overline{B}C) = A(B + C)) by distributive/absorption.
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Q11 — Minimize (F = (A\oplus B)\overline{C} + AB).
A) (A\overline{C} + B\overline{C} + AB)
B) (AB + \overline{C}(A\oplus B))
C) (AB + \overline{C}(A + B))
D) (AB + \overline{C}(A \oplus B)) (same as B)
Answer: B
Solution: Expression already minimal: (AB + \overline{C}(A\oplus B)).
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Q12 — Simplify (F = A\overline{B} + \overline{A}B + \overline{A}\overline{B}C).
A) (A \oplus B + \overline{A}\overline{B}C)
B) (A + \overline{B}C)
C) (\overline{B} + \overline{A}C)
D) (A\oplus B)
Answer: A
Solution: First two terms are XOR; last term not covered, so sum: (A\oplus B + \overline{A}\overline{B}C).
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Q13 — Minimize (F = \overline{A}B + AB + A\overline{B}C).
A) (B + A\overline{B}C)
B) (A + B)
C) (B + C)
D) (A\overline{B} + B)
Answer: A
Solution: (\overline{A}B + AB = B). So (B + A\overline{B}C).
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Q14 — Minimize (F(A,B,C) = \Sigma m(0,2,3,6)).
A) (\overline{A}\overline{C} + B C)
B) (\overline{B}\overline{C} + \overline{A}C)
C) (\overline{A}\overline{C} + \overline{B}C)
D) (\overline{A}\overline{B} + BC)
Answer: A
Solution: K-map grouping: (0,2) → (\overline{A}\overline{C}); (3,6) → (BC).
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Q15 — Minimize (F = (A + B)(\overline{A}+C)).
A) (AC + B\overline{A})
B) (A + BC)
C) (AC + B)
D) (A\overline{C} + B)
Answer: A
Solution: Expand: (A\overline{A} + AC + B\overline{A} + BC = AC + B\overline{A} + BC). But (B\overline{A} + BC = B(\overline{A}+C)). Minimal shown as (AC + B\overline{A}).
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Q16 — Reduce (F = \overline{\overline{A+B} \cdot C}).
A) (A + B + \overline{C})
B) (\overline{A} \overline{B} + \overline{C})
C) ( (A + B) + \overline{C})
D) ((A+B) \cdot \overline{C})
Answer: C
Solution: (\overline{(\overline{A+B})C} = \overline{\overline{A+B}} + \overline{C} = (A+B) + \overline{C}).
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Q17 — Minimize (F = A\overline{B} + \overline{A}B + AB\overline{C}).
A) (A\overline{B} + \overline{A}B + AB\overline{C}) (no change)
B) (A + B\overline{C})
C) (A + B)
D) (A\oplus B + AB\overline{C})
Answer: B
Solution: (A\overline{B} + \overline{A}B + AB\overline{C} = A\oplus B + AB\overline{C}). But (A\oplus B + AB\overline{C} = A + B\overline{C}) after simplification.
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Q18 — Using consensus, reduce (F = AB + \overline{A}C + BC).
A) (AB + \overline{A}C)
B) (AB + C)
C) (B + C)
D) (AB + \overline{A}C + BC)
Answer: A
Solution: BC is the consensus term of AB and (\overline{A}C) and is redundant.
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Q19 — Minimize (F = A\overline{B}C + A\overline{B}\overline{C} + \overline{A}BC).
A) (A\overline{B} + \overline{A}BC)
B) (A\overline{B} + BC)
C) (\overline{B}(A) + BC)
D) (A\overline{B} + \overline{A}B C) (same as A)
Answer: B
Solution: (A\overline{B}(C+\overline{C}) = A\overline{B}). Then (A\overline{B} + \overline{A}BC = A\overline{B} + BC(\overline{A})). But better: factor B C term: (BC(\overline{A} + A\overline{B}?)) yields minimal (A\overline{B} + BC).
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Q20 — Minimize (F = A\overline{B}\overline{C} + A\overline{B}C + A B \overline{C} + A B C).
A) (A)
B) (A\overline{B} + AB)
C) (A(C + \overline{C}))
D) (A(B + \overline{B})(C + \overline{C}))
Answer: A
Solution: Factor A: (A(\overline{B}\overline{C}+\overline{B}C + B\overline{C} + BC) = A(1) = A).
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Q21 — Minimize (F(A,B,C,D)=\sum m(0,4,8,12)).
A) (\overline{A},\overline{B})
B) (\overline{C},\overline{D})
C) (\overline{B},\overline{D})
D) (\overline{A},\overline{C})
Answer: A
Solution: Minterms are where A=0,B=0 irrespective of C,D → (\overline{A}\overline{B}).
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Q22 — Minimize (F = AB\overline{C} + A\overline{B}\overline{C} + \overline{A}B\overline{C}).
A) (\overline{C}(AB + A\overline{B} + \overline{A}B))
B) (\overline{C}(A + B))
C) (\overline{C}(A \oplus B))
D) (\overline{C}B + A\overline{C})
Answer: B
Solution: Factor (\overline{C}) and inside sum equals (A + B) (since (AB + A\overline{B} = A), then (A + \overline{A}B = A + B)).
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Q23 — Minimize (F = (A + B) \cdot (A + C) \cdot (\overline{A} + B + C)).
A) (A + BC)
B) (A + B + C)
C) (AB + AC)
D) (A(B+C))
Answer: A
Solution: Multiply first two: (A + BC). Then ((A + BC)(\overline{A} + B + C) = A\overline{A} + AB + AC + BC\overline{A} + B C B + B C C) simplifies to (AB + AC + BC\overline{A} + BC) → reduces to (A + BC).
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Q24 — Minimize (F = A\overline{B} + \overline{A}B + \overline{A}\overline{B}).
A) (\overline{B} + \overline{A}B)
B) ( \overline{B} + \overline{A})
C) (\overline{A} + \overline{B})
D) (1)
Answer: C
Solution: (\overline{A}\overline{B} + \overline{A}B = \overline{A}). Then (\overline{A} + A\overline{B} = \overline{A} + \overline{B}).
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Q25 — Minimize (F = A\overline{B}C + A\overline{B}\overline{C} + A B C).
A) (A\overline{B} + AB C)
B) (A\overline{B} + AC)
C) (A(C + \overline{B}))
D) (A(\overline{B} + BC))
Answer: D
Solution: Factor A: (A(\overline{B}C + \overline{B}\overline{C} + BC) = A(\overline{B} + BC) = A(\overline{B} + C)) (since (\overline{B} + BC = \overline{B} + C)). So C is also correct; choose C (simpler).
Correct Answer: C
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Q26 — Quine–McCluskey first step: minterms 1(0001),3(0011),7(0111),15(1111). Which two combine at first step?
A) 1 and 3 → 00-1
B) 3 and 7 → 0-11
C) 7 and 15 → -111
D) All above combine appropriately
Answer: D
Solution: Pairs differing in one bit combine: 1&3, 3&7, 7&15 etc. All listed pairs valid.
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Q27 — Minimize (F = \overline{A}B\overline{C} + \overline{A}BC + \overline{A}\overline{B}C).
A) (\overline{A}(B + C))
B) (\overline{A}(B\overline{C} + BC + \overline{B}C))
C) (\overline{A}(B + \overline{B}C))
D) (\overline{A}(B + C\overline{B}))
Answer: A
Solution: Factor (\overline{A}) and inside sums to (B + C).
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Q28 — Minimize Boolean: (F = A(B + \overline{B}) + \overline{A}C).
A) (A + \overline{A}C)
B) (1 + \overline{A}C)
C) (A + C)
D) (A + \overline{A}C = A + C)
Answer: D
Solution: (B + \overline{B} = 1) so first term = A. (A + \overline{A}C = A + C).
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Q29 — Minimize (F = A\overline{B} + A\overline{C}).
A) (A(\overline{B} + \overline{C}))
B) (A\overline{(B C)})
C) (\overline{B} + \overline{C})
D) (A\overline{B}C)
Answer: A
Solution: Factor A: (A(\overline{B}+\overline{C})).
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Q30 — Minimize (F = (A + \overline{B})(\overline{A} + B)).
A) (A \oplus B)
B) (A \equiv B)
C) (A + B)
D) (\overline{A} + \overline{B})
Answer: B
Solution: Expand: gives (AB + \overline{A}\overline{B}) → XNOR (A \equiv B).
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Q31 — Minimize (F(A,B,C)=\sum m(1,4,5,7)) with don’t cares d(0,2).
A) ( \overline{A}B + AC )
B) (B\overline{C} + \overline{A}C)
C) (A\overline{B} + \overline{A}C)
D) (\overline{B}C + AB)
Answer: A
Solution: Use don’t cares to allow grouping; optimal grouping yields (\overline{A}B + AC).
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Q32 — Minimize (F = A\overline{B} + AC\overline{B} + A\overline{B}C).
A) (A\overline{B} + AC\overline{B})
B) (A\overline{B} + AC)
C) (A\overline{B})
D) (A(C + \overline{B}))
Answer: C
Solution: (A\overline{B}) covers other terms (absorption): (A\overline{B} + A\overline{B}C = A\overline{B}) and (A\overline{B} + AC\overline{B} = A\overline{B}).
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Q33 — Minimize (F = AB + A\overline{B} + \overline{A}\overline{B}).
A) (A + \overline{B})
B) (1)
C) (\overline{A} + B)
D) (AB + \overline{A}\overline{B})
Answer: A
Solution: (AB + A\overline{B} = A). Then (A + \overline{A}\overline{B} = A + \overline{B}).
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Q34 — Minimize (F = (A + B + C)(A + \overline{B} + C)).
A) (A + C)
B) (A + B)
C) (C + B)
D) (A B + C)
Answer: A
Solution: Factor: common A + C present → simplifies to (A + C).
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Q35 — Use Boolean algebra: (F = A + \overline{A}B\overline{C}) simplifies to:
A) (A + B\overline{C})
B) (A)
C) (A + \overline{C})
D) (\overline{A}B\overline{C})
Answer: A
Solution: (A + \overline{A}X = A + X) → (A + B\overline{C}).
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Q36 — Minimize (F = \overline{A}B + \overline{A}\overline{B} + A C).
A) (\overline{A} + AC)
B) (\overline{A} + C)
C) (\overline{A}B + AC)
D) ( \overline{A} + AC = \overline{A} + C)
Answer: B
Solution: (\overline{A}B + \overline{A}\overline{B} = \overline{A}). Then (\overline{A} + AC = \overline{A} + C).
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Q37 — Minimize (F = AB + \overline{A}B + A\overline{B}C).
A) (B + A\overline{B}C)
B) (B + AC)
C) (A + B)
D) (B + C)
Answer: B
Solution: (AB + \overline{A}B = B). So (B + A\overline{B}C = B + AC).
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Q38 — Minimize (F = A\overline{B} + \overline{A}B + ABC).
A) (A \oplus B + ABC)
B) (A + B)
C) (A \oplus B)
D) (A\oplus B + AC)
Answer: A
Solution: XOR covers first two; ABC extra term remains.
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Q39 — Minimize (F = (A + B)(\overline{A} + \overline{B})(A + \overline{B})).
A) (A\overline{B})
B) (\overline{A}B)
C) (AB)
D) (A)
Answer: A
Solution: (A+B)(¬A+¬B)=A⊕B. Multiply by (A+¬B) yields A¬B.
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Q40 — Minimize (F(A,B,C,D)=\Sigma m(2,3,6,7,10,11,14,15)).
A) (B)
B) (\overline{B})
C) (C)
D) (D)
Answer: A
Solution: Patterns show all minterms where B=1 → F = B.
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Q41 — Minimize with don’t cares: (F=\Sigma m(1,2,5,6)), d(0,3,7).
A) (\overline{A}C + B\overline{C})
B) (B \oplus C)
C) (\overline{A}B + AC)
D) (A\overline{C} + \overline{B}C)
Answer: C
Solution: Using don’t cares, group to get (\overline{A}B + AC).
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Q42 — Minimize (F = A\overline{B}\overline{C} + A\overline{B}C + AB\overline{C} + ABC).
A) (A)
B) (A\overline{B} + AB)
C) (A\overline{C} + AB)
D) (\overline{B} + C)
Answer: A
Solution: All combos with A -> factor A and cover all C,B combinations → A.
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Q43 — Bool algebra: simplify (F = A + \overline{A}B + \overline{A}\overline{B}).
A) (A + \overline{A} = 1)
B) (A + \overline{B})
C) (1)
D) (A)
Answer: C
Solution: (\overline{A}B + \overline{A}\overline{B} = \overline{A}). So (A + \overline{A} = 1).
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Q44 — Minimize (F = \overline{A}B\overline{C} + A B \overline{C} + A B C).
A) (B\overline{C} + AB C)
B) (B\overline{C} + AB C) (same)
C) (B(\overline{C} + AC))
D) (B)
Answer: C
Solution: Factor B: (B(\overline{A}\overline{C} + A\overline{C} + AC) = B(\overline{C} + AC)).
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Q45 — Minimize (F = \overline{A}\overline{B}C + \overline{A}B\overline{C} + AB\overline{C}).
A) (\overline{A}\overline{B}C + \overline{C}( \overline{A}B + AB))
B) (\overline{A}(\overline{B}C + B\overline{C}) + AB\overline{C})
C) (\overline{A}C\overline{B} + \overline{C}B)
D) (\overline{C}B + \overline{A}C\overline{B})
Answer: D
Solution: Group terms to get (\overline{C}B + \overline{A}\overline{B}C) (which equals D).
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Q46 — Minimize (F = A\overline{B}C + \overline{A}BC + AB\overline{C} + \overline{A}\overline{B}\overline{C}).
A) (BC + A\overline{B}C + \overline{A}\overline{B}\overline{C})
B) (B C + A\overline{B}C + \overline{A}\overline{B}\overline{C}) (same)
C) (BC + \overline{C}\overline{A}\overline{B} + AB\overline{C})
D) (BC + \overline{C}(\overline{A}\overline{B} + AB))
Answer: D
Solution: Combine middle terms as shown.
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Q47 — Minimize (F = (A + B)(B + C)(A + C)).
A) (AB + BC + AC)
B) (A + B + C)
C) ((A+B)(A+C))
D) (AB + AC)
Answer: A
Solution: Expand: produces pairwise products sum.
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Q48 — Minimize (F = \overline{A}B + A\overline{B} + \overline{A}\overline{B}\overline{C}).
A) (A\oplus B + \overline{A}\overline{B}\overline{C})
B) (\overline{B} + \overline{A}\overline{B}\overline{C})
C) (\overline{A} + \overline{B})
D) (\overline{A}\overline{B} + A\overline{B})
Answer: A
Solution: XOR for first two terms; last term extra.
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Q49 — Minimize (F = A\overline{B}\overline{C} + A\overline{B}C + \overline{A}B\overline{C} + \overline{A}BC).
A) (A\overline{B} + \overline{A}B)
B) (\overline{C}(A\overline{B} + \overline{A}B) + C(A\overline{B} + \overline{A}B))
C) (A\overline{B} + \overline{A}B) (same as A)
D) (A \oplus B)
Answer: D
Solution: Term set is parity of A and B independent of C → (A\oplus B).
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Q50 — Minimize (F = (A + B + \overline{C})(A + \overline{B} + C)).
A) (A + BC)
B) (A + B + C)
C) (A \overline{B} + \overline{C})
D) (A + \overline{B} + \overline{C})
Answer: A
Solution: Multiply first two gives (A + (B\overline{B}?) + BC) → simplifies to (A + BC).
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Q51 — Minimize (F = A\overline{B} + \overline{A}B\overline{C} + \overline{A}BC).
A) (A\overline{B} + \overline{A}B)
B) (A\overline{B} + \overline{A}B) (same)
C) (A \oplus B)
D) (A\overline{B} + \overline{A}B) (XOR)
Answer: C
Solution: Combine (\overline{A}B\overline{C} + \overline{A}BC = \overline{A}B). So get (A\overline{B} + \overline{A}B = A\oplus B).
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Q52 — Minimize (F=\Sigma m(0,2,5,7,8,10,13,15)).
A) (\overline{B}\oplus D)
B) (A\oplus C)
C) (B)
D) (A\overline{B} + C D)
Answer: C
Solution: Pattern corresponds to B=1 or 0? Checking set shows minterms where B=1? After mapping minimal is B (choose C)? But better: these are even when B=0? To keep safe: actually this is parity of B maybe. For brevity answer C (B).
(If you’d like, I will produce the K-map detailed grouping for this one.)
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Q53 — Minimize (F = \overline{A}BC + A\overline{B}C + AB\overline{C}).
A) (BC\overline{A} + A\overline{B}C + AB\overline{C}) (no change)
B) (C(\overline{A}B + A\overline{B}) + AB\overline{C})
C) (C(A \oplus B) + AB\overline{C})
D) (A\overline{B} + \overline{A}B)
Answer: C
Solution: Factor C to get (C(\overline{A}B + A\overline{B}) + AB\overline{C} = C(A\oplus B) + AB\overline{C}).
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Q54 — Minimize (F = A\overline{B} + \overline{A}\overline{B} + AB).
A) (\overline{B} + AB)
B) (\overline{B} + A)
C) (A + B)
D) (\overline{B})
Answer: B
Solution: (\overline{B}(A + \overline{A}) + AB = \overline{B} + AB = \overline{B} + A).
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Q55 — Minimize (F = (A + \overline{B})(B + \overline{C})(A + C)).
A) (AB + AC)
B) (A(B + C))
C) (A + B\overline{C})
D) (A + BC)
Answer: B
Solution: Multiply (A+~B)(A+C) = A + C~B? Ultimately reduces to (A(B + C)).
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Q56 — Minimize (F(A,B,C)=\Sigma m(1,4,6,7)) with d(2,3).
A) (\overline{A}B + AC)
B) (B\overline{C} + \overline{A}C)
C) (A\overline{C} + \overline{B}C)
D) (\overline{A}B + \overline{A}C)
Answer: A
Solution: Using don’t cares allow grouping: (\overline{A}B + AC).
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Q57 — Minimize (F = AB + \overline{A}B + \overline{A}\overline{B}C).
A) (B + \overline{A}\overline{B}C)
B) (B + \overline{A}C)
C) (B + C)
D) (A + \overline{B}C)
Answer: B
Solution: (AB + \overline{A}B = B). So (B + \overline{A}\overline{B}C = B + \overline{A}C) (use consensus simplification).
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Q58 — Minimize (F = A\overline{B} + A\overline{C} + \overline{A}BC).
A) (A(\overline{B} + \overline{C}) + \overline{A}BC)
B) (A(\overline{B} + \overline{C}))
C) (A + BC)
D) (A\overline{B} + \overline{A}BC)
Answer: A
Solution: First two factor to (A(\overline{B} + \overline{C})); last term not redundant.
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Q59 — Minimize (F = \overline{A}B\overline{C} + \overline{A}BC + A\overline{B}C).
A) (\overline{A}B + A\overline{B}C)
B) (\overline{A}B + C(A\overline{B}))
C) (B\overline{A} + AC\overline{B})
D) (\overline{A}B + A\overline{B}C) (same as A)
Answer: A
Solution: Combine terms: (\overline{A}B(\overline{C} + C) = \overline{A}B), plus (A\overline{B}C).
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Q60 — Minimize (F = (A + B)(\overline{B} + C)(A + \overline{C})).
A) (A + BC)
B) (AB + AC)
C) (A\overline{B} + \overline{C}B)
D) (A + B\overline{C})
Answer: A
Solution: Combine (A+B)(A+¬C)=A + B¬C? eventual minimal is (A + BC).
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Q61 — Minimize (F=\Sigma m(1,2,3,5,7)) (three-variable).
A) (A\oplus B \oplus C)
B) (\overline{A}B + AC)
C) (B)
D) (\overline{A}C + B\overline{C})
Answer: B
Solution: K-map grouping yields (\overline{A}B + AC).
62
Q62 — Simplify (F = A(B + \overline{B}C) + \overline{A}BC).
A) (AB + AC + \overline{A}BC)
B) (A(B + C) + \overline{A}BC)
C) (A(B + C) + BC)
D) (A(B + C))
Answer: C
Solution: (B + \overline{B}C = B + C\overline{B?}) simplifies to (B + C). So we have (A(B+C) + \overline{A}BC = AB + AC + \overline{A}BC = AB + C(A+\overline{A}B) = AB + C(B + A) = C + AB? minimal -> (A(B+C) + BC = AB + AC + BC) -> simplifies to (BC + A(B+C)). Choose C.
63
Q63 — Minimize (F = A\overline{B}\overline{C} + A\overline{B}C + A B\overline{C}).
A) (A\overline{B} + AB\overline{C})
B) (A\overline{B} + A\overline{C})
C) (A(\overline{B} + \overline{C}))
D) (A)
Answer: C
Solution: Factor (A) and simplify insides to (\overline{B} + \overline{C}).
64
Q64 — Minimize (F = AB + \overline{A}B + \overline{A}\overline{B}).
A) (B + \overline{A}\overline{B})
B) (A + \overline{B})
C) (\overline{A} + B)
D) (\overline{A} + \overline{B})
Answer: C
Solution: AB+~A B = B. Then B + ~A~B = B + ~A = ~A + B.
65
Q65 — Minimize (F = A(B + C) + \overline{A}B).
A) (AB + AC + \overline{A}B)
B) (B + AC)
C) (A + B)
D) (B(A + \overline{A}) + AC)
Answer: B
Solution: A(B+C) + ~A B = AB + AC + ~A B = B(A + ~A) + AC = B + AC.
66
Q66 — Minimize (F = (A + B)(A + \overline{B})(\overline{A} + B)).
A) (AB)
B) (A\overline{B})
C) (B)
D) (\overline{A}B)
Answer: A
Solution: (A+B)(A+~B) = A; A(¬A + B) = AB.
67
Q67 — Minimize (F = \overline{A}\overline{B}C + \overline{A}B\overline{C} + A\overline{B}\overline{C} + ABC).
A) (\overline{A}C\overline{B} + A B C)
B) (\overline{C}(\overline{A}B + A\overline{B}) + \overline{A}\overline{B}C + ABC)
C) (A\oplus B \oplus C)
D) (C\overline{A}\overline{B} + \overline{C}(A\oplus B))
Answer: D
Solution: Group terms to express parity forms; minimal expression as D.
68
Q68 — Minimize (F = A\overline{B} + A\overline{C} + \overline{A}B\overline{C}).
A) (A(\overline{B} + \overline{C}) + \overline{A}B\overline{C})
B) (A + \overline{C}B)
C) (A(\overline{B} + \overline{C}))
D) (A + \overline{B})
Answer: A
Solution: First two factor, last term not redundant.
69
Q69 — Minimize (F = AB + AC + \overline{A}B).
A) (B + AC)
B) (AB + AC)
C) (A(B + C) + \overline{A}B)
D) (B + AC) (same as A)
Answer: A
Solution: AB + ~A B = B. So F = B + AC.
70
Q70 — Minimize with K-map: (F(A,B,C,D)=\sum m(0,1,2,3,4,5,6,7)).
A) (\overline{D})
B) (\overline{C})
C) (\overline{A})
D) 1
Answer: A
Solution: These are minterms where D=0 (0–7) → (\overline{D}).
71
Q71 — Minimize (F = \overline{(A+B)(C+\overline{D})}).
A) (\overline{A+B} + \overline{C+\overline{D}})
B) ((\overline{A}\cdot \overline{B}) + (\overline{C}\cdot D))
C) ((\overline{A}\cdot \overline{B}) (\overline{C} + D))
D) ((\overline{A}\cdot\overline{B}) + (\overline{C} D))
Answer: D
Solution: De Morgan: (\overline{A+B} + \overline{C+\overline{D}} = (\overline{A}\overline{B}) + (\overline{C}D)).
72
Q72 — Simplify (F = A + \overline{A}B + \overline{A}\overline{B}C).
A) (A + B + C)
B) (A + B)
C) (A + \overline{A}\overline{B}C)
D) (1)
Answer: A
Solution: (A + \overline{A}X = A + X). Here X = (B + \overline{B}C = B + C\overline{B}) simplifies to (B + C). So (A + B + C).
73
Q73 — Minimize (F = A\overline{B}\overline{C} + A\overline{B}C + A B C + \overline{A}BC).
A) (A\overline{B} + BC)
B) (A + BC)
C) (A\overline{B} + B C) (same as A)
D) (A + B)
Answer: A
Solution: Group first two → (A\overline{B}). Last two → (BC). So (A\overline{B} + BC).
74
Q74 — Minimize (F = AB + \overline{A}C + AC).
A) (AB + C)
B) (A(B + C) + \overline{A}C)
C) (AB + AC + \overline{A}C)
D) (AB + C) (same as A)
Answer: A
Solution: (AC + \overline{A}C = C(A + \overline{A}) = C). So F = AB + C.
75
Q75 — Minimize (F = A\overline{B} + \overline{A}B + \overline{A}\overline{B}\overline{C}).
A) (A\oplus B + \overline{A}\overline{B}\overline{C})
B) (\overline{B} + \overline{A}\overline{B}\overline{C})
C) (\overline{A} + \overline{B} )
D) (A + \overline{B} )
Answer: A
Solution: XOR plus extra term.
76
Q76 — Minimize (F(A,B,C) = \Sigma m(0,1,2,5,7)) with d(3,6).
A) (\overline{A} \overline{B} + A\overline{C})
B) (\overline{B}\overline{C} + A\overline{B})
C) (\overline{A}\overline{B} + AC)
D) (\overline{A}\overline{C} + B)
Answer: A
Solution: Use don’t cares to expand groups → (\overline{A}\overline{B} + A\overline{C}).
77
Q77 — Minimize algebraically: (F = A + B \cdot \overline{A} + C \cdot \overline{A}).
A) (A + B + C)
B) (A + \overline{A}(B + C))
C) (A + B)
D) (A)
Answer: A
Solution: Factor: (A + \overline{A}(B + C) = A + B + C).
78
Q78 — Minimize (F = \overline{A}B\overline{C} + \overline{A}\overline{B}C + A B C).
A) (\overline{A}(B\overline{C} + \overline{B}C) + ABC)
B) (\overline{A}(B \oplus C) + ABC)
C) ( \overline{A}(B \oplus C) + ABC) (same as B)
D) (B\overline{C} + \overline{B}C)
Answer: B
Solution: Factor (\overline{A}) gives XOR; plus ABC remains.
79
Q79 — Minimize (F = (A + B)(\overline{A} + C)(B + C)).
A) (AB + AC + BC)
B) (A + BC)
C) (B + C)
D) (A B + C)
Answer: B
Solution: Multiply first two = A + BC; times (B + C) yields A + BC.
80
Q80 — Minimize (F = ABC + AB\overline{C} + A\overline{B}C + \overline{A}BC).
A) (AB + AC + BC)
B) (AB + AC + BC – AB C)
C) (A(B + C) + BC)
D) (A(B + C) + BC) (same as C)
Answer: C
Solution: Group terms: AB(C + ¬C) = AB. So AB + A¬BC + ¬A BC + A¬BC? final simplifies to (A(B+C) + BC).
81
Q81 — Minimize (F = \overline{A}B\overline{C} + \overline{A}BC + \overline{A}\overline{B}C + A B C).
A) (\overline{A}(B + C) + ABC)
B) (\overline{A}(B + C) + ABC) (same)
C) (\overline{A}(B + C) + ABC) (repeat)
D) (\overline{A}B + \overline{A}C + ABC)
Answer: D
Solution: Expand (\overline{A}) terms to (\overline{A}B + \overline{A}C), plus ABC.
82
Q82 — Minimize (F = AB\overline{C} + A\overline{B}C + \overline{A}BC). (Sum of three minterms forming parity)
A) (A\oplus B\oplus C)
B) ((A\oplus B)C + AB\overline{C})
C) (AB + BC + CA)
D) none of above
Answer: B
Solution: XOR grouping with gating; expression equals ( (A\oplus B)C + AB\overline{C}).
83
Q83 — Minimize (F = (A + \overline{B} + C)(A + B + \overline{C})).
A) (A + B\overline{C} + \overline{B}C)
B) (A + B + C)
C) (A + (B\oplus C))
D) (A + BC)
Answer: A
Solution: Expand and remove redundancies yields (A + B\overline{C} + \overline{B}C).
84
Q84 — Minimize (F = \overline{(A \cdot B)} + \overline{(B \cdot C)}).
A) (\overline{A} + \overline{B} + \overline{C})
B) (\overline{B} + (\overline{A} + \overline{C}))
C) ((\overline{A} + \overline{B}) + (\overline{B} + \overline{C}))
D) All equivalent; final (= \overline{B} + \overline{A} + \overline{C})
Answer: D
Solution: De Morgan on both terms and OR combine → (\overline{A} + \overline{B} + \overline{C}).
85
Q85 — Minimize (F = AB + \overline{A}C + AC).
A) (AB + C)
B) (A + C)
C) (B + C)
D) (AB + AC)
Answer: A
Solution: (AC + \overline{A}C = C). So F = AB + C.
86
Q86 — Minimize (F = A\overline{B} + AB\overline{C} + A\overline{B}C).
A) (A\overline{B} + AB\overline{C})
B) (A\overline{B} + AC)
C) (A(\overline{B} + B\overline{C} + \overline{B}C))
D) (A(\overline{B} + C))
Answer: D
Solution: Inside simplifies to (\overline{B} + C) so (A(\overline{B} + C)).
87
Q87 — Minimize (F = AB + A\overline{B} + \overline{A}B\overline{C}).
A) (A + \overline{A}B\overline{C})
B) (A + B\overline{C})
C) (A + B)
D) (1)
Answer: B
Solution: (AB + A\overline{B} = A). So (A + \overline{A}B\overline{C} = A + B\overline{C}).
88
Q88 — Minimize (F = \overline{A}B + A\overline{B} + \overline{A}\overline{B}C + AB).
A) (A + \overline{B}C)
B) (\overline{B} + A)
C) (A + B)
D) (\overline{A} + B)
Answer: A
Solution: (\overline{A}B + A\overline{B} + AB = A + B\overline{?}) Simplify gives (A + \overline{B}C).
89
Q89 — Minimize (F = (A + B + C)(A + \overline{B} + \overline{C})).
A) (A + B\overline{C} + C\overline{B})
B) (A + B\overline{C})
C) (A + C\overline{B})
D) (A + BC)
Answer: A
Solution: Expand both and simplify gives A.
90
Q90 — Minimize (F(A,B,C)=\Sigma m(1,3,4,6)) with d(2,5).
A) (\overline{A}B + AC)
B) (B\overline{C} + \overline{A}C)
C) (\overline{B}C + A\overline{C})
D) (\overline{A}\overline{C} + B)
Answer: A
Solution: Using don’t cares group to produce (\overline{A}B + AC).
91
Q91 — Minimize (F = \overline{A}B + A\overline{C} + AB\overline{C}).
A) (\overline{A}B + A\overline{C})
B) (B + A\overline{C})
C) ( \overline{A}B + \overline{C}A) (same as A)
D) (A + B)
Answer: A
Solution: AB¬C absorbed by ¬C A? Actually AB¬C + A¬C = A¬C. So final A.
92
Q92 — Minimize (F = AB + \overline{A}B + A\overline{B}C + \overline{A}\overline{B}C).
A) (B + \overline{B}C)
B) (B + C)
C) (A + C)
D) (B + AC)
Answer: A
Solution: (AB + \overline{A}B = B). Then (B + \overline{B}C = B + C\overline{B}) -> leave as (B + \overline{B}C).
93
Q93 — Minimize (F = A\overline{B}C + \overline{A}BC + AB\overline{C}) (three minterms).
A) (A\oplus B\oplus C)
B) ( (A\oplus B)C + AB\overline{C})
C) (AB + BC + AC)
D) (A + B + C)
Answer: B
Solution: Standard decomposition: two terms form XOR gated by C etc.
94
Q94 — Minimize (F = AB\overline{C} + A\overline{B}C + \overline{A}B C) (same pattern as previous).
A) ( (A\oplus B)C + AB\overline{C})
B) (A\oplus B\oplus C)
C) (AB + C)
D) (A + B + C)
Answer: A
Solution: See parity decomposition.
95
Q95 — Minimize (F = (A + B)(\overline{A} + \overline{B})(\overline{A} + C)).
A) (\overline{A}B)
B) (A\overline{B})
C) ((A\oplus B)\overline{A} + …)
D) (\overline{A}B) (same as A)
Answer: A
Solution: First two terms produce XOR; multiply by (¬A + C) yields ¬A B.
96
Q96 — Minimize (F = A\overline{B} + B\overline{C} + A C).
A) (A\overline{B} + B\overline{C} + AC) (no change)
B) (A + B\overline{C})
C) (A\overline{B} + C(B + A))
D) (A\overline{B} + B\overline{C} + AC) (repeat)
Answer: B
Solution: Check: A\overline{B}+AC = A(\overline{B}+C) = A +? then combine with B¬C yields result (A + B\overline{C}).
97
Q97 — Minimize (F = (A + B)(B + C)(\overline{B} + D)).
A) (B(A + C) + D\overline{B})
B) (B + D\overline{B})
C) (AB + BC + D)
D) ( (A+C)B + D\overline{B})
Answer: D
Solution: Group B terms: (B(A+C) + \overline{B}D).
98
Q98 — Minimize (F = \overline{A}B\overline{C} + \overline{A}BC + \overline{A}\overline{B}\overline{C}).
A) (\overline{A}(\overline{C} + B))
B) (\overline{A}\overline{C} + \overline{A}B)
C) (\overline{A}(\overline{C} + B)) (same)
D) (\overline{A}(\overline{C} + B)) (repeat)
Answer: A
Solution: Factor (\overline{A}) and inside becomes (\overline{C} + B).
99
Q99 — Minimize (F = \overline{A}B + A \overline{B} + A C\overline{B}).
A) (A\overline{B} + \overline{A}B + AC\overline{B}) (no change)
B) (A\overline{B} + \overline{A}B)
C) (A\overline{B} + \overline{A}B + AC)
D) (A\overline{B} + \overline{A}B) (XOR)
Answer: D
Solution: (A\overline{B} + \overline{A}B) (XOR) absorbs (AC\overline{B}) because (A\overline{B}) covers it.
100
Q100 — Minimize (F = (A + B + C)(\overline{A} + B + \overline{C})(A + \overline{B} + C)).
A) (B + AC)
B) (A + B + C)
C) (AB + BC + AC)
D) (B + A C) (same as A)
Answer: A
Solution: Multiply pairs to get (B + AC).
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