---
title: "Tautology, contradiction, logical equivalence"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:tautology-contradiction-equivalence"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/tautology-contradiction-equivalence
site: BCT Engineering
---

# Tautology, contradiction, logical equivalence

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2075 Bhadra, 3+2 marks
- Video: [Tautology, Contradiction, Contingency & Satisfiability](https://www.youtube.com/watch?v=yo7aULLUz_0) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/tautology-contradiction-equivalence
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Define the terms Tautology, Contradiction and Logical Equivalences. Show that ¬A → ¬B and B → A are Logically Equivalent.

## 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.

## Answer

#### 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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A tautology is a proposition that is?
   A. True for at least one assignment
   B. True for every assignment
   C. False for every assignment
   D. Neither true nor false
   Answer: B. A tautology is true under all assignments, e.g. p ∨ ¬p. (Key point: Tautology)

2. Which is a contradiction?
   A. p ∨ ¬p
   B. p → p
   C. p ∧ ¬p
   D. p ∨ p
   Answer: C. p ∧ ¬p is false for every value of p. (Key point: Contradiction)

3. A ≡ B means?
   A. A and B are true for some common assignment
   B. A → B is true once
   C. A and B have the same truth value in every case (A ↔ B is a tautology)
   D. A and B have the same number of variables
   Answer: C. Equivalence requires identical truth values in every row. (Key point: Logical equivalence)

4. The algebraic proof of ¬A → ¬B ≡ B → A uses which steps?
   A. De Morgan only
   B. Absorption and distribution
   C. Implication law, double negation, commutativity, implication law
   D. Modus ponens
   Answer: C. ¬A → ¬B ≡ A ∨ ¬B ≡ ¬B ∨ A ≡ B → A. (Key point: Algebraic proof)

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