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

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

Option (B) is the correct negation: "It rain and I shall not go to school."

Step 1: Understanding the Negation of a Conditional

A conditional statement "If P, then Q" is logically equivalent to "P implies Q." The negation of a conditional statement is "P and not Q."

Step 2: Identify Components of the Original Statement

The original statement is "If it rains, I shall go to school." Here, P = "It rains" and Q = "I shall go to school." The negation of this conditional is thus "It rains and I shall not go to school."

Step 3: Match the Components with the Options

We need to find which option correctly represents "It rains and I shall not go to school." Looking at the options, option (B) matches "It rain and I shall not go to school," making it the negation of the given conditional statement.

Key Concepts

Conditional Statements

In logical reasoning, a conditional statement is an if-then statement that links two propositions. It is often expressed as "If P, then Q." Here, P is the antecedent, and Q is the consequent.
Conditional statements are found everywhere in daily language and mathematics. For example, "If it rains, then I will take my umbrella." This indicates that the action of taking an umbrella is contingent upon it raining.

• Antecedent (P): The condition or premise; "it rains" in our example.
• Consequent (Q): The result or outcome; "I will take my umbrella" in the given statement.

Understanding these statements is vital as they form the basis of logical arguments, helping you reason effectively in both everyday situations and academic exercises.

Negation

Negation is the process of making a statement false if it is true, and true if it is false. In the context of conditional statements, this involves converting the original statement into its opposite meaning.
To negate a conditional "If P, then Q," we use "P and not Q." This form shows that while the condition (P) holds, the result (Q) does not follow. In the exercise example:

• Original: "If it rains, I shall go to school."
• Negation: "It rains and I shall not go to school."

This negation changes the logical relationship by making it clear that even if P occurs, Q will not. This is a fundamental aspect of understanding contradictions.

Logical Equivalence

Logical equivalence occurs when two statements are true in the same scenarios and false in the same scenarios. For conditional statements, knowing the logical equivalences can simplify complex logical reasoning.
For the conditional "If P, then Q," remember:

• It is logically equivalent to "Not P or Q." This means that if P is false, or Q is true, the conditional is valid.
• Its negation "P and not Q" does not share logical equivalence with the original, but it's crucial to identify contrasting outcomes.

Recognizing equivalent statements is essential for problem-solving, allowing you to transform and manipulate logical expressions effectively.