---
title: "Truth value of a compound formula (b)"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:truth-value-biconditional-formula"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/truth-value-biconditional-formula
site: BCT Engineering
---

# Truth value of a compound formula (b)

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Bhadra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/truth-value-biconditional-formula
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

If P = F, Q = T, S = T, R = F, then find truth value of: ((P ∧ ¬Q) ↔ (Q ∧ R)) → (S ∨ ¬Q)

## Short answer

((P∧¬Q)↔(Q∧R)) is true (F↔F), and (S∨¬Q) is true, so the implication is true. Truth value: TRUE.

## Answer

Given P = F, Q = T, S = T, R = F. Formula: ((P ∧ ¬Q) ↔ (Q ∧ R)) → (S ∨ ¬Q)

| Step | Sub-formula | Working | Value |
|---|---|---|---|
| 1 | ¬Q | ¬T | F |
| 2 | P ∧ ¬Q | F ∧ F | F |
| 3 | Q ∧ R | T ∧ F | F |
| 4 | (P ∧ ¬Q) ↔ (Q ∧ R) | F ↔ F | T |
| 5 | S ∨ ¬Q | T ∨ F | T |
| 6 | whole: step 4 → step 5 | T → T | T |

**Answer: the truth value is TRUE.**

## Check yourself

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

1. With P = F, Q = T, what is P ∧ ¬Q?
   A. T
   B. F
   C. Depends on R
   D. Undefined
   Answer: B. P is false, so the conjunction is false. (Key point: P ∧ ¬Q)

2. With P ∧ ¬Q = F and Q ∧ R = F, what is (P ∧ ¬Q) ↔ (Q ∧ R)?
   A. F
   B. T
   C. Depends on S
   D. Undefined
   Answer: B. A biconditional is true when both sides have the same value. (Key point: Biconditional)

3. With S = T, Q = T, S ∨ ¬Q is?
   A. F
   B. T
   C. Undefined
   D. Same as ¬Q
   Answer: B. T ∨ F = T. (Key point: S ∨ ¬Q)

4. The whole implication T → T is?
   A. FALSE
   B. TRUE
   C. Undefined
   D. Depends on R
   Answer: B. T → T is true. (Key point: Final value)

## Close variants set in other exams

- [Truth value of a compound formula (a)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Atruth-value-s-pr-formula) — Discrete Structure, chapter 1

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