Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Truth value of the statement " \(p\) or \(q\) " is false, when (A) \(p\) is true, \(q\) is false (B) \(p\) is false, \(q\) is true (C) \(p\) and \(q\) both are true (D) \(p\) and \(q\) both are false

Step-by-Step Solution

Answer

Option (D) is correct: both \( p \) and \( q \) are false.

Step 1: Understand the 'or' Statement in Logic

An 'or' statement, denoted as \( p \lor q \), in logic is true if at least one of the components, \( p \) or \( q \), is true. It is only false if both \( p \) and \( q \) are false.

Step 2: Analyze Option (A)

In option (A), \( p \) is true and \( q \) is false. Using the rule from Step 1, \( p \lor q \) is true because \( p \) is true. Therefore, this option does not satisfy the condition where the statement is false.

Step 3: Analyze Option (B)

In option (B), \( p \) is false and \( q \) is true. According to the 'or' operation rule, \( p \lor q \) is true because \( q \) is true. Hence, this option does not result in a false statement.

Step 4: Analyze Option (C)

In option (C), both \( p \) and \( q \) are true. An 'or' statement is true if at least one is true, so \( p \lor q \) is true. Thus, this option does not make the statement false.

Step 5: Analyze Option (D)

In option (D), both \( p \) and \( q \) are false. Since both components are false, \( p \lor q \) becomes false. This fits the condition where the 'or' statement is false.

Step 6: Conclusion

After analyzing all options, the statement \( p \lor q \) is false only in option (D) when both \( p \) and \( q \) are false.

Key Concepts

Truth Table

A truth table is a helpful tool in logic. It's used to visually display all possible truth values of logical statements.

For a statement involving logical operators like "or," the truth table shows how the truth or falsity of the complete statement changes. The truth table consists of rows of truth values for every possible combination of the truth values of the variables involved.

• For our exercise, we have two variables: \( p \) and \( q \), which can each be true (T) or false (F).
• The truth table for the "or" statement \( p \lor q \) would have four possible rows: (TT, TF, FT, FF).

In logical terms, a statement \( p \lor q \) is only false when both \( p \) and \( q \) are false. This rule is crucial for understanding which combinations of \( p \) and \( q \) make \( p \lor q \) false, which in the given exercise is option (D).

Disjunction in Logic

Disjunction is a logical operation that connects two statements with "or." In logic, it's represented as \( \lor \) and written in a formula as \( p \lor q \).
Disjunctions are true if either of the individual statements is true.

• If \( p = \text{True} \textbf{ and/or } q = \text{True} \), then \( p \lor q \) becomes True, as only one or both need to be true.
• The statement only evaluates to False when both \( p \) and \( q \) are false.

This concept of disjunction is important for logical reasoning as it helps in constructing valid arguments and determining the veracity of statements. In our task, when both \( p \) and \( q \) are false, the disjunction fails, making it false only in option (D).

Logical Reasoning

Logical reasoning refers to the process of using a structured, methodologically sound approach to arrive at a conclusion based on given premises. In logic exercises, like the one presented, it requires understanding logical operations and applying the appropriate rules.
Logical reasoning involves:

• Recognizing how operators like "or" work in different scenarios.
• Assessing each possibility carefully to determine the truth value.

Using logical reasoning, we correctly analyzed each option A through D by checking the truth values of \( p \) and \( q \). Eventually, determining that option (D) was the only condition making the statement false. This approach not only arrives at the correct answer but deepens the understanding of logical operations within statement analysis.