Problem 56 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider the following two statements \(\mathrm{P}:\) If 7 is an odd number, then 7 is divisible by \(2 .\) \(Q\) : If 7 is a prime number, then 7 is an odd number. If \(\mathrm{V}_{1}\) is the truth value of the contrapositive of \(\mathrm{P}\) and \(\mathrm{V}_{2}\) is the truth value of contrapositive of \(Q\), then the ordered pair \(\left(\mathrm{V}_{1}, \mathrm{~V}_{2}\right)\) equals: (a) \((\mathrm{F}, \mathrm{F})\) (b) \((\mathrm{F}, \mathrm{T})\) (c) \((\mathrm{T}, \mathrm{F})\) (d) \((\mathrm{T}, \mathrm{T})\)

Step-by-Step Solution

Answer

The ordered pair is (F, T).

Step 1: Understand Statement P

The statement \( P \) is "If 7 is an odd number, then 7 is divisible by 2." A conditional statement \( A \rightarrow B \) is false if \( A \) is true and \( B \) is false. Here, the antecedent '7 is an odd number' is true, but the consequent '7 is divisible by 2' is false. Therefore, \( P \) is false.

Step 2: Find the Contrapositive of P

The contrapositive of a statement \( A \rightarrow B \) is \( eg B \rightarrow eg A \). For \( P \), the contrapositive is "If 7 is not divisible by 2, then 7 is not an odd number." Since '7 is not divisible by 2' is true and '7 is not an odd number' is false, the contrapositive is false.

Step 3: Understand Statement Q

The statement \( Q \) is "If 7 is a prime number, then 7 is an odd number." Here the antecedent '7 is a prime number' is true, and the consequent '7 is an odd number' is also true. Therefore, \( Q \) is true.

Step 4: Find the Contrapositive of Q

The contrapositive of a statement \( A \rightarrow B \) is \( eg B \rightarrow eg A \). For \( Q \), the contrapositive is "If 7 is not an odd number, then 7 is not a prime number." Since '7 is not an odd number' is false, the statement is true because a false antecedent in an implication makes the implication true.

Step 5: Evaluate Ordered Pair (V1, V2)

From Steps 2 and 4, we know that the contrapositive of \( P \) is false, thus \( V_1 = F \), and the contrapositive of \( Q \) is true, hence \( V_2 = T \). Therefore, the ordered pair \( (V_1, V_2) \) is \( (F, T) \).

Key Concepts

Contrapositive

In mathematical logic, the contrapositive is a critical concept that helps us understand conditional statements better. A conditional statement is typically of the form "If A, then B", noted as \( A \rightarrow B \). The contrapositive of this statement switches and negates both parts: "If not B, then not A", noted as \( eg B \rightarrow eg A \). This transformation often helps simplify problems and reveal truths about logical statements.

For example, consider the statement "If it rains, the ground gets wet." The contrapositive would be "If the ground is not wet, then it did not rain."

A crucial aspect of contrapositives is that they always have the same truth value as the original statement. This means if the original statement is true, its contrapositive is also true, and if the original statement is false, its contrapositive is also false. Understanding this symmetry helps in determining the correctness of logical arguments in mathematics.

Truth Values

In logic, each statement has a truth value indicating whether it is true or false. Understanding the truth value of contrapositives and other statements is essential in assessing logical outcomes.

When you evaluate a statement like \'If A, then B\' (\( A \rightarrow B \)), you check both \( A \) (the antecedent) and \( B \) (the consequent) to determine the truth condition. A rule in logic states that a conditional statement is false only when \( A \) is true, and \( B \) is false. In all other cases—whether \( A \) is false, \( B \) is true, or both—the conditional statement is true. This rule helps in analyzing problems logically and deducing accurate conclusions.

• True Antecedent, False Consequent: False Conditional
• False Antecedent, Any Consequent: True Conditional
• True Antecedent, True Consequent: True Conditional

This logic rule forms the backbone for evaluating both regular conditionals and their contrapositives.

Conditional Statements

Conditional statements form the basis for much of logical reasoning in mathematics. Expressed generally as "If A, then B" (\( A \rightarrow B \)), these statements connect two propositions and define a relationship between them.

Understanding how to read and analyze conditionals is crucial for identifying logical structures and their consequences. For example, consider the statement "If 7 is a prime number, then 7 is an odd number." This implies a relationship where the truth of '7 is prime' leads to the truth of '7 is odd.'

To further analyze conditional statements, you must be aware of various related forms:

• **Converse**: Switches the place of A and B ("If B, then A")
• **Inverse**: Negates both A and B ("If not A, then not B")
• **Contrapositive**: Switches and negates both A and B ("If not B, then not A")

Learning these variations allows for a deeper investigation into logic and the ability to judge the truth-value relationships effectively, especially as applied in mathematical problem-solving.