- 2075 Bhadra · 3+2 marks
Tautology, contradiction, logical equivalence
Short answer
A tautology is always true (e.g., p ∨ ¬p); a contradiction always false (p ∧ ¬p). Logical equivalence means identical truth values. ¬A→¬B ≡ B→A: truth tables are identical, or algebraically ¬A→¬B ≡ A ∨ ¬B ≡ ¬B ∨ A ≡ B→A.
Define the terms Tautology, Contradiction and Logical Equivalences. Show that ¬A → ¬B and B → A are Logically Equivalent.
Tautology
A compound proposition that is true for every assignment of truth values to its variables. Example: p ∨ ¬p.
Contradiction
A compound proposition that is false for every assignment. Example: p ∧ ¬p.
Logical equivalence
Two propositions A and B are logically equivalent (A ≡ B) if they have the same truth value in every case, that is, A ↔ B is a tautology.
Show ¬A → ¬B ≡ B → A
Truth table:
| A | B | ¬A | ¬B | ¬A → ¬B | B → A |
|---|---|---|---|---|---|
| T | T | F | F | T | T |
| T | F | F | T | T | T |
| F | T | T | F | F | F |
| F | F | T | T | T | T |
The columns for ¬A → ¬B and B → A are identical in all four rows. Hence ¬A → ¬B ≡ B → A. ∎
Algebraic check. ¬A → ¬B ≡ ¬(¬A) ∨ ¬B (implication law) ≡ A ∨ ¬B (double negation) ≡ ¬B ∨ A (commutative) ≡ B → A (implication law).
✓ Answer checked by Claude Opus 5.5 (high effort)
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