Problem 61 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The negation of the statement "If I become a teacher, then I will open a school", is: (a) I will become a teacher and I will not open a school. (b) Either I will not become a teacher or I will not open a school. (c) Neither I will become a teacher nor I will open a school. (d) I will not become a teacher or I will open a school.

Step-by-Step Solution

Answer

(a) I will become a teacher and I will not open a school.

Step 1: Understanding Conditional Statements

The original statement is a conditional statement: "If I become a teacher, then I will open a school." In symbolic logic, this is represented as \( p \to q \), where \( p \) is "I become a teacher" and \( q \) is "I will open a school." The aim is to find the negation of this statement.

Step 2: Identifying the Negation of Conditional Statements

The negation of a conditional statement \( p \to q \) is \( p \land eg q \). This can be understood as asserting that the first part is true (\( p \)) and the second part is false (\( eg q \)). Thus, the negation is saying "I become a teacher and I will not open a school."

Step 3: Matching with Provided Options

Now, we match our derived negation \( p \land eg q \) with the given options. (a) "I will become a teacher and I will not open a school." perfectly matches \( p \land eg q \). Thus, option (a) is the correct answer.

Key Concepts

Conditional Statements

In logic, a conditional statement is an assertion that communicates a dependency between two conditions or events. It's structured in the format "If... then..." and is used to express that the truth of one statement (the consequent) relies on the truth of another (the antecedent). Consider the example conditional statement: "If I become a teacher, then I will open a school." Here, the antecedent is "I become a teacher," denoted by \( p \), and the consequent is "I will open a school," represented as \( q \). Together, this is written in symbolic logic as \( p \to q \).

Conditional statements are central in logical reasoning because they help us deduce outcomes based on given premises. They set a foundation for understanding implications in logic, computer science, and everyday decision-making processes. However, understanding the negation of these statements opens the door to deeper logical exploration.

Negation in Logic

Negation in logic involves contradicting or denying a certain statement. For conditional statements like \( p \to q \), the negation is not directly \( eg(p \to q) \) but rather \( p \land eg q \). This means that the first part \( p \) is true, but the consequent \( q \) does not happen. Think of it as portraying a situation where the premise occurs, but the expected outcome doesn't follow.

Using our previous example, the negation "If I become a teacher, then I will not open a school" can be depicted in symbolic logic as "I become a teacher and I will not open a school." This focuses on invalidating the original conditional connection that doing one thing leads to another, offering a critical tool to test assumptions and understand logical validity without relying solely on intuition.

Symbolic Logic

Symbolic logic is a powerful branch of logic that uses symbols and variables to represent logical expressions and arguments. It simplifies complex propositions and facilitates easier manipulation and analysis of these statements. In the exercise example, "If I become a teacher, then I will open a school" is symbolically expressed as \( p \to q \).

In symbolic logic:

• \( p \) stands for a proposition or statement (e.g., "I become a teacher").
• \( q \) refers to another proposition (e.g., "I will open a school").
• \( p \to q \) denotes that when \( p \) happens, \( q \) follows.

Negation (\( eg \)) is used to depict the opposite of a given statement in symbolic expressions. This precise language aids in breaking down the statements into their logical components, enabling a clearer understanding of their relationships and implications in diverse fields such as mathematics, computer science, and philosophy.