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

# Truth value of a compound formula (a)

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2070 Bhadra, 4 marks
- Video: [THE TRUTH VALUE OF A COMPOUND STATEMENT](https://www.youtube.com/watch?v=Mks4yjKTkwo) — MATHStorya
- 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-s-pr-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: (S → (P ∧ ¬R)) ∧ ((P → (R ∨ Q)) ∧ S)

## Short answer

S→(P∧¬R) evaluates to false (T→F), while ((P→(R∨Q))∧S is true. The conjunction is false; truth value: FALSE.

## Answer

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

#### Part 1: S → (P ∧ ¬R)
- ¬R = ¬F = T
- P ∧ ¬R = F ∧ T = F
- S → (P ∧ ¬R) = T → F = **F**

#### Part 2: (P → (R ∨ Q)) ∧ S
- R ∨ Q = F ∨ T = T
- P → (R ∨ Q) = F → T = T
- (P → (R ∨ Q)) ∧ S = T ∧ T = **T**

#### Whole formula
F ∧ T = F

| Sub-formula | Value |
|---|---|
| ¬R | T |
| P ∧ ¬R | F |
| S → (P ∧ ¬R) | F |
| R ∨ Q | T |
| P → (R ∨ Q) | T |
| (P → (R ∨ Q)) ∧ S | T |
| Whole formula | F |

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

## Check yourself

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

1. With P = F and R = F, what is P ∧ ¬R?
   A. T
   B. F
   C. Undefined
   D. Cannot tell
   Answer: B. ¬R = T and P ∧ T = F since P is false. (Key point: Evaluating P ∧ ¬R)

2. With S = T and P ∧ ¬R = F, S → (P ∧ ¬R) is?
   A. T
   B. F
   C. T only if Q is T
   D. Undefined
   Answer: B. T → F is false. (Key point: Implication)

3. What is (P → (R ∨ Q)) ∧ S?
   A. F
   B. T
   C. Depends on S only
   D. Undefined
   Answer: B. R ∨ Q = T, P → T = T, and T ∧ S(T) = T. (Key point: Second part)

4. The whole formula is?
   A. TRUE
   B. FALSE
   C. Depends on Q
   D. Cannot be determined
   Answer: B. F ∧ T = F. (Key point: Final value)

## Close variants set in other exams

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

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