Problem 62 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p\) and \(q\) denote the following statements \(p:\) The sun is shining \(q\) : I shall play tennis in the afternoon The negation of the statement "If the sun is shining then I shall play tennis in the afternoon", is Online (a) \(q \Rightarrow \sim p\) (b) \(q \wedge \sim p\) (c) \(p \wedge \sim q\) (d) \(\sim q \Rightarrow-p\)

Step-by-Step Solution

Answer

The correct answer is (c) \(p \wedge \sim q\).

Step 1: Identify the Given Conditional Statement

The statement we have is "If the sun is shining then I shall play tennis in the afternoon", which is expressed in logical terms as \( p \rightarrow q \). This is a conditional statement that links two propositions: \( p \) (the sun is shining) and \( q \) (I shall play tennis in the afternoon).

Step 2: Understand the Negation of a Conditional Statement

The negation of a conditional statement \( p \rightarrow q \) is not \( p \rightarrow q \). Instead, it is expressed as \( p \wedge \sim q \), meaning that the first part is true (the sun is shining), but the second part is false (I shall not play tennis).

Step 3: Apply the Negation to the Statement

Apply the negation rule to the given conditional statement \( p \rightarrow q \). The negation will be \( p \wedge \sim q \), which reads as "The sun is shining and I shall not play tennis in the afternoon".

Step 4: Identify the Correct Option

Given the choices, the correct negation corresponds to option (c), which is \( p \wedge \sim q \). This matches the negation we found in the previous step.

Key Concepts

Conditional Statements

Conditional statements are a fundamental part of propositional logic, serving as the building blocks for creating logical relationships between propositions. In logic, a conditional statement is typically written in the form “if...then...”, and is represented symbolically as \( p \rightarrow q \). Here, \( p \) is the hypothesis or the antecedent, and \( q \) is the conclusion or consequent. The statement implies that if \( p \) is true, then \( q \) must also be true.
For example, the conditional statement "If the sun is shining, then I shall play tennis in the afternoon" can be represented as \( p \rightarrow q \), where \( p \) represents "the sun is shining" and \( q \) represents "I shall play tennis in the afternoon". An important aspect of conditional statements is understanding that if the hypothesis \( p \) is false, the entire statement \( p \rightarrow q \) is considered true, regardless of the truth value of \( q \). This counterintuitive rule helps in maintaining consistency within logical systems.
Exploring further, breaking down complex relationships into simple conditional statements is a practical way to analyze logical arguments or make decisions based on multiple criteria.

Logical Propositions

Logical propositions are declarative sentences that can either be true or false. They form the basic units of propositional logic. Each statement or proposition is represented by variables typically denoted by letters such as \( p \), \( q \), \( r \), and so on.
For a sentence to qualify as a logical proposition, it must assert a clear fact or condition. For example, "The sun is shining" is a proposition denoted by \( p \), which can independently be evaluated as true or false. Similarly, "I shall play tennis in the afternoon" can be seen as a proposition \( q \).
Understanding that propositions have fixed truth values allows us to use them in logical operations and connect them with other propositions to create compound statements. This foundational understanding is vital as we move towards the use of logical connectors like \( \wedge \) (and), \( \vee \) (or), and \( \rightarrow \) (implies), which combine individual propositions into more complex logical statements.

Propositional Logic

Propositional logic is a branch of logic that deals with propositions and their relationships using logical connectives. It provides a framework for analyzing the truth of propositions in a structured and mathematical manner.
In propositional logic, propositions are manipulated using connectives such as:

• Conjunction (\(\wedge\)): This connects two propositions into one, stating that both must be true. For instance, \( p \wedge q \) is true only when both \( p \) and \( q \) are true.
• Disjunction (\(\vee\)): This allows either one or both propositions to be true. \( p \vee q \) is true when at least one of \( p \) or \( q \) is true.
• Negation (\(\sim\)): This flips the truth value of a proposition. If \( p \) is true, \( \sim p \) is false.
• Implication (\(\rightarrow\)): This forms a conditional relationship, as discussed in conditional statements.

By understanding these basic connectives and their impact on the truth values of combined propositions, one can solve complex logical problems or construct proofs.
Additionally, propositional logic forms the basis for more advanced areas of logic and computer science, such as boolean algebra and the development of algorithms. It is an essential tool for anyone looking to master logical reasoning and problem-solving skills.