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

# Tableau for (p∧¬q)→s

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2071 Bhadra, 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-p-and-not-q-implies-s
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

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

## Answer

Formula Φ = (p ∧ ¬q) → s.

**Rules.** A → B is a β-formula and splits into ¬A | B. Then ¬(p ∧ ¬q) is a β-formula and splits into ¬p | ¬¬q, and ¬¬q is replaced by q.

#### Tableau
```text
            (p ∧ ¬q) → s
              /        \
      ¬(p ∧ ¬q)          s
        /     \           o
      ¬p      ¬¬q
       o       |
               q
               o
```

| Branch | Literals | Status |
|---|---|---|
| 1 | ¬p | open |
| 2 | q | open |
| 3 | s | open |

All three branches are open (no complementary pair), so Φ is **satisfiable**. The models are: p = F; or q = T; or s = T.

## Check yourself

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

1. (p ∧ ¬q) → s first splits into?
   A. p ∧ ¬q | s
   B. ¬(p ∧ ¬q) | s
   C. ¬p | s
   D. p | ¬q | s
   Answer: B. A → B splits into ¬A | B. (Key point: Root split)

2. ¬(p ∧ ¬q) is expanded as?
   A. ¬p and q on one branch
   B. ¬p | ¬¬q (then q)
   C. p | q
   D. ¬p | ¬q
   Answer: B. ¬(A ∧ B) splits into ¬A | ¬B, and ¬¬q becomes q. (Key point: Second split)

3. How many open branches, and what does it show?
   A. None; unsatisfiable
   B. One; valid
   C. Three open branches; satisfiable
   D. Two closed; unsatisfiable
   Answer: C. Branches ¬p, q and s are all open, so the formula is satisfiable. (Key point: Result)

## 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)∨r)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Atableau-not-pq-or-r) — 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×