Match List I with List II by identifying the logically equivalent…
2025
Match List I with List II by identifying the logically equivalent propositions.
List I: Proposition | List II: Equivalent form |
|---|---|
A. p→q | I. (p∧q)∨(¬p∧¬q) |
B. ¬(p∨(¬p∧q)) | II. ¬p∨q |
C. p↔q | III. ¬(p∨q) |
D. ¬(p↔q) | IV. ¬p↔q |
Answer: C. A-II, B-III, C-I, D-IV — ConceptLogical equivalence means that two propositions have the same truth value for every assignment of their variables. Standard identities include p→q ≡…
- A.
A-I, B-III, C-II, D-IV
- B.
A-II, B-II, C-III, D-IV
- C.
A-II, B-III, C-I, D-IV
- D.
A-II, B-III, C-IV, D-I
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Show answer & explanation
Correct answer: C
Concept
Logical equivalence means that two propositions have the same truth value for every assignment of their variables. Standard identities include p→q ≡ ¬p∨q and p↔q ≡ (p∧q)∨(¬p∧¬q).
De Morgan’s laws and distributivity simplify compound propositions, while negating a biconditional produces exclusive-or: ¬(p↔q) ≡ (p∧¬q)∨(¬p∧q) ≡ ¬p↔q.
Application
For A, use the implication identity: p→q ≡ ¬p∨q. Therefore A matches II.
For B, distribute inside the negation: p∨(¬p∧q) ≡ (p∨¬p)∧(p∨q) ≡ True∧(p∨q) ≡ p∨q. Hence ¬(p∨(¬p∧q)) ≡ ¬(p∨q), so B matches III.
For C, use the biconditional identity: p↔q ≡ (p∧q)∨(¬p∧¬q). Therefore C matches I.
For D, negate the biconditional: ¬(p↔q) ≡ (p∧¬q)∨(¬p∧q). This is also ¬p↔q, so D matches IV.
Cross-check
The table cross-checks the three standard identity pairs for A, C, and D across all four assignments of p and q; the distributive derivation in the application independently verifies B.
p | q | p→q / ¬p∨q | p↔q / (p∧q)∨(¬p∧¬q) | ¬(p↔q) / ¬p↔q |
|---|---|---|---|---|
T | T | T / T | T / T | F / F |
T | F | F / F | F / F | T / T |
F | T | T / T | F / F | T / T |
F | F | T / T | T / T | F / F |
Result
The complete matching is A–II, B–III, C–I, D–IV.
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