---
title: "Satisfiable and unsatisfiable formulas"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:define-satisfiable-unsatisfiable"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/define-satisfiable-unsatisfiable
site: BCT Engineering
---

# Satisfiable and unsatisfiable formulas

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Magh, 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/define-satisfiable-unsatisfiable
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Define satisfiable and unsatisfiable formulas.

## Short answer

A satisfiable formula has at least one interpretation (truth assignment) that makes it true. An unsatisfiable formula (contradiction) is false for every possible interpretation.

## Answer

- **Satisfiable formula.** A propositional formula is satisfiable if there is at least one interpretation (assignment of truth values to its variables) that makes it true. Such an interpretation is called a model. Example: p ∨ q is satisfiable (p = T makes it true).
- **Unsatisfiable formula.** A formula is unsatisfiable (a contradiction) if it is false under every interpretation. Example: p ∧ ¬p.

Relations: a valid formula (tautology) is true under all interpretations, so it is satisfiable. A formula A is valid exactly when ¬A is unsatisfiable.

## Check yourself

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

1. A formula is satisfiable if?
   A. It is true under every interpretation
   B. It is true under at least one interpretation
   C. It is false under every interpretation
   D. It has no variables
   Answer: B. At least one model exists. (Key point: Satisfiable)

2. Which formula is unsatisfiable?
   A. p ∨ q
   B. p ∧ ¬p
   C. p → q
   D. p ∨ ¬p
   Answer: B. p ∧ ¬p is false under every interpretation. (Key point: Unsatisfiable)

3. A is valid exactly when?
   A. A is unsatisfiable
   B. ¬A is unsatisfiable
   C. ¬A is satisfiable
   D. A is a contradiction
   Answer: B. If ¬A is never true, A is always true. (Key point: Validity relation)