---
title: "Tableau for ¬((p∧q)∨r)"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:tableau-not-pq-or-r"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/tableau-not-pq-or-r
site: BCT Engineering
---

# Tableau for ¬((p∧q)∨r)

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Magh, 3 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/tableau-not-pq-or-r
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Draw the tableau for the formula Φ = ¬((p∧q)∨r), where ¬ denotes the negation of a variable, ∨ denotes the disjunction of variables and ∧ denotes the conjunction of variables.

## Answer

Formula Φ = ¬((p ∧ q) ∨ r).

**Rules.** ¬(A ∨ B) is an α-formula and gives ¬A and ¬B on the same branch. ¬(p ∧ q) is a β-formula and splits into ¬p | ¬q.

#### Tableau
```text
          ¬((p ∧ q) ∨ r)
                |
           ¬(p ∧ q)
                |
               ¬r
              /    \
            ¬p      ¬q
             o       o
```

| Branch | Literals | Status |
|---|---|---|
| Left | ¬r, ¬p | open |
| Right | ¬r, ¬q | open |

Both branches are open, so Φ is **satisfiable**. Models: r = F with p = F, or r = F with q = F. (Φ is false whenever r is true, or when p and q are both true.)

## Check yourself

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

1. ¬((p ∧ q) ∨ r) gives which on the same branch?
   A. p ∧ q and r
   B. ¬(p ∧ q) | ¬r
   C. ¬p and ¬q and ¬r
   D. ¬(p ∧ q) and ¬r
   Answer: D. ¬(A ∨ B) is an α-formula giving ¬A and ¬B. (Key point: First rule)

2. ¬(p ∧ q) is expanded as?
   A. ¬p and ¬q on one branch
   B. ¬p | ¬q
   C. p | q
   D. ¬p only
   Answer: B. ¬(A ∧ B) is a β-formula. (Key point: Branching)

3. Both branches are open. What does this mean?
   A. Φ is unsatisfiable
   B. Φ is satisfiable
   C. Φ is a tautology
   D. Nothing
   Answer: B. An open fully expanded branch gives a model. (Key point: Outcome)

## Close variants set in other exams

- [Tableau for a formula set](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Atableau-formula-set) — Discrete Structure, chapter 1, asked 5×
- [Tableau for (p∧¬q)→s](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Atableau-p-and-not-q-implies-s) — Discrete Structure, chapter 1, asked 5×
- [Tableau for (T∨S)→¬Q](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Atableau-t-or-s-implies-not-q) — Discrete Structure, chapter 1, asked 5×