Match the Boolean-algebra laws to their relational forms: (A) Distributive law, (B) De Morgan’s law, (C) Idempotent law, (D) Involution law — with (1) x + x = x, (2) x(y + z) = xy + xz, (3) (x)̄̄ = x, (4) (x + y)̄ = x̄ ȳ.
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AA-1, B-2, C-3, D-4
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BA-2, B-4, C-1, D-3
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CA-3, B-4, C-1, D-2
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DA-4, B-1, C-2, D-3
Answer
Correct Answer: A-2, B-4, C-1, D-3
Explanation
Introduction:Boolean algebra laws provide the algebraic toolkit for simplifying logic expressions and designing digital circuits. This item tests recognition of four cornerstone identities and their canonical forms.
Given Data / Assumptions:
- A: Distributive law.
- B: De Morgan’s law.
- C: Idempotent law.
- D: Involution (double-complement) law.
- Relational forms: (1) x + x = x, (2) x(y + z) = xy + xz, (3) (x)̄̄ = x, (4) (x + y)̄ = x̄ ȳ.
Concept / Approach:
Each law has a standard, widely used expression. The distributive law expands products over sums. De Morgan’s laws transform complements of sums/products. Idempotence states redundancy of combining a variable with itself. Involution states that complementing twice restores the original variable.
Step-by-Step Solution:
A → (2): x(y + z) = xy + xz (distribution of product over sum).B → (4): (x + y)̄ = x̄ ȳ (the dual is (xy)̄ = x̄ + ȳ).C → (1): x + x = x (and x·x = x).D → (3): (x)̄̄ = x (double negation).Verification / Alternative check:
Truth tables or Venn diagrams confirm each identity; Karnaugh maps also illustrate idempotence and distributivity.
Why Other Options Are Wrong:
Any permutation that mismatches these canonical forms contradicts standard Boolean identities taught in digital logic.
Common Pitfalls:
Confusing De Morgan’s transformations or mixing the two De Morgan forms; forgetting that idempotence applies to both + and · operations.
Final Answer:
A-2, B-4, C-1, D-3