Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Negation of the conditional: "If it rains, I shall go to school" is (A) It rains and I shall go to school. (B) It rain and I shall not go to school. (C) It does not rain and I shall go to school. (D) None of these

Step-by-Step Solution

Answer

(B) It rain and I shall not go to school.

Step 1: Understanding the Conditional Statement

The given conditional statement is "If it rains, I shall go to school." In logical form, a conditional statement can be written as \( p \rightarrow q \), meaning "if \( p \), then \( q \)." Here, \( p \) = "it rains" and \( q \) = "I shall go to school."

Step 2: Identifying the Negation of the Conditional

The negation of a conditional statement \( p \rightarrow q \) is \( p \land eg q \), which means \( p \) is true and \( q \) is false. This can be translated to "It rains and I shall not go to school."

Step 3: Matching the Negation to Options

From the explanation, the correct negation of "If it rains, I shall go to school" is "It rains and I shall not go to school." This matches with option (B).

Key Concepts

Conditional Statements

Conditional statements are a fundamental building block in mathematics and logic. They follow the "if-then" structure, written as "if \( p \), then \( q \)." Here, \( p \) is the hypothesis or condition, and \( q \) is the conclusion or result. In this structure, if the hypothesis \( p \) is true, then the conclusion \( q \) is expected to be true as well. However, if the hypothesis is false, the truth of the conclusion does not affect the truth of the statement as a whole.
In the example "If it rains, I shall go to school," the statement follows the conditional logic. "It rains" is the hypothesis \( p \), and "I shall go to school" is the conclusion \( q \). This means that rain acts as a condition under which going to school will occur. Understanding how these statements are constructed helps in further exploring their meanings and transformations.

Negation

Negation, within the context of logic and mathematics, is about finding the opposite or contradiction of a given statement. When dealing with conditional statements, negation requires flipping the statement's meaning to signal a situation where the original statement's conclusion does not occur, even if the condition does.
For a conditional statement like \( p \rightarrow q \), its negation is represented as \( p \land eg q \). This translates into logical terms as "\( p \) is true and \( eg q \) is true." In our example, this becomes: "It rains and I shall not go to school." Here, the negation involves not only recognizing when the initial condition \( p \) (it rains) happens but also ensuring the result \( q \) (I shall go to school) does not follow. Understanding this change is important for correctly interpreting logical situations.

Logical Reasoning

Logical reasoning is the process that allows us to form conclusions based on a set of premises or conditions. It's an essential skill in mathematics and everyday problem-solving, guiding how we interpret information and make decisions.
When analyzing conditional statements and their negations, logical reasoning helps determine the validity of the arguments presented. In our problem, reasoning through the options and understanding both the original and negated statements leads us to the correct choice among the given options.

• Carefully analyze the given statement and identify the components \( p \) and \( q \).
• Understand that a conditional statement's negation involves confirming the truth of the hypothesis and negation of the conclusion simultaneously.
• Evaluate the provided options, using your reasoning skills to match the correct logical interpretation of the negation.

By employing logical reasoning, students can more efficiently identify correct logical transformations and solve similar exercises with confidence.