Problem 23 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \((\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r}) \rightarrow \sim p \vee q\) is false, then the truth values of \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) are respectively \(\quad\) [Online April 15, 2018] (a) \(\mathrm{F}, \mathrm{T}, \mathrm{F}\) (b) \(\mathrm{T}, \mathrm{F}_{2} \mathrm{~T}\) \(\begin{array}{ll}\text { (c) } \mathrm{F}, \mathrm{F}, \mathrm{F} & \text { (d) } \mathrm{T}, \mathrm{T}, \mathrm{T}\end{array}\)

Step-by-Step Solution

Answer

The truth values are (b) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\).

Step 1: Understanding the Implication

We need to determine the circumstances under which the expression \((\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r}) \rightarrow \sim p \vee q\) is false. An implication \(A \rightarrow B\) is false only when \(A\) is true, and \(B\) is false.

Step 2: Simplifying the Antecedent

The antecedent is \((\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\). This simplifies to \(\mathrm{p} \wedge \sim \mathrm{q} \wedge \mathrm{p} \wedge \mathrm{r}\), which further simplifies to \(\mathrm{p} \wedge \sim \mathrm{q} \wedge \mathrm{r}\). For this antecedent to be true, \(\mathrm{p}\) must be true, \(\mathrm{q}\) must be false, and \(\mathrm{r}\) must be true.

Step 3: Simplifying the Consequent

The consequent is \(\sim p \vee q\). This is false only when both \(\sim p\) (or \(p\)) is false and \(q\) is false.

Step 4: Determining Consistent Values for Variables

Combining both requirements from Step 2 and Step 3: \(\mathrm{p} = \text{True}\), \(\mathrm{q} = \text{False}\) (because \(\sim q\) must be true), \(\mathrm{r} = \text{True}\) (from the antecedent), and \(\mathrm{p}\) is true from both implications. This matches with option (b) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\).

Key Concepts

Logical Connectives

Logical connectives are the building blocks of propositional logic, allowing us to form complex propositions from simpler ones. They act like the operators in arithmetic but are used to combine truth values rather than numbers.

Some common logical connectives include:

• AND (\( \wedge \)): This connective is true only when both connected statements are true. It represents the idea of conjunction.
• OR (\( \vee \)): This is true if at least one of the connected statements is true, capturing the essence of disjunction.
• NOT (\( \sim \)): This operator inverts the truth value of a statement. If a proposition is true, \( \sim \text{proposition} \) makes it false, and vice versa.
• IMPLIES (\( \rightarrow \)): This is false only when the first statement (antecedent) is true and the second statement (consequent) is false. Otherwise, it is true.

Understanding these connectives is crucial for forming and interpreting logical statements, as they dictate how the truth values of individual propositions affect the truth values of compound statements.

Truth Values

In propositional logic, propositions are statements that can either be true or false, but not both. These truth values form the basis of logical computations and analyses.

Let's break down the truth values guiding this logic system:

• True (T): This label is given to a proposition if it accurately describes the state of affairs. It's equivalent to saying "the statement is valid."
• False (F): This reflects a proposition that does not correspond to the truth, meaning the statement is invalid.

When dealing with compound propositions formed using logical connectives, evaluating truth values involves checking individual truth conditions outlined by each connective. For example:
- In a conjunctive statement (\( p \wedge q \)), both \( p \) and \( q \) must be true for the statement to hold true.
Effectively understanding how these truth values interact within connectives guides logical reasoning and problem-solving tasks.

Implication in Logic

Implication, symbolized as \( \rightarrow \), is a fundamental concept in logic that describes a conditional relationship between statements. It reads as "if...then," formalizing the concept of one statement leading to another.

Here's a detailed look at how implication works:

• Structure: An implication is structured as \( p \rightarrow q \), where \( p \) is the antecedent and \( q \) is the consequent.
• Truth Conditions: The implication is true except when the antecedent \( p \) is true, and the consequent \( q \) is false. In all other scenarios, \( p \rightarrow q \) is considered true.
• Example: Consider "If it is raining, then the ground is wet." This statement is only false if it is raining, yet the ground is not wet. If either or both conditions are not met, the implication holds as true.

Grasping the truth conditions of implications is vital in logic as it helps distinguish between possible outcomes in logical reasoning, enabling one to deduce valid conclusions from given premises.