Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(p \rightarrow(p \wedge \sim q)\) is false, then the truth values of \(p\) and \(q\) are respectively: (a) \(\mathrm{F}_{1} \mathrm{~F}\) (b) \(\mathrm{T}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{T}\) (d) \(\mathrm{F}, \mathrm{T}\)

Step-by-Step Solution

Answer

(c) T, T

Step 1: Understand the Logical Expression

The expression given is \(p \rightarrow (p \wedge \sim q)\). It denotes implication, where \(p\) implies the conjunction of \(p\) and \(\sim q)\).

Step 2: Interpret Implication and Conjunction

The implication \(p \rightarrow (p \wedge \sim q)\) is false only when \(p\) is true and \((p \wedge \sim q)\) is false. In implications, \(A \rightarrow B\) is false only when \(A\) is true and \(B\) is false.

Step 3: Determine when \(p \wedge \sim q\) is False

\(p \wedge \sim q\) is false when either \(p\) is false or \(\sim q\) is false. However, since \(p\) must be true for the implication to be false, \(\sim q\) must be false, meaning \(q\) must be true.

Step 4: Verify Conditions

If \(p\) is true and \(q\) is true, then \(\sim q\) is false, making \(p \wedge \sim q\) false. Thus, \(p \rightarrow (p \wedge \sim q)\) is false only when \(p\) is true and \(q\) is true.

Key Concepts

Implication

In mathematical logic, an implication is a fundamental concept denoted by the symbol \(\rightarrow\). This symbol connects two statements, usually referred to as the antecedent and the consequent. You can think of an implication as a kind of promise that if the first part (the antecedent) is true, then the second part (the consequent) has to be true as well.
Implications are often seen in the form \(p \rightarrow q\). There is only one situation where this implication is false: when \(p\) is true and \(q\) is false. So, if you promise something with an implication and don't deliver, it results in a false statement. In every other scenario:

• When both \(p\) and \(q\) are true
• When both \(p\) and \(q\) are false
• When \(p\) is false and \(q\) is true

The implication will be true, even if it seems counterintuitive in some cases (like promising something but not needing to follow through when you never have to act on it because \(p\) is false). This idea of implication is crucial in logical reasoning and proofs.

Conjunction

Conjunction is another key concept in mathematical logic, represented by the symbol \(\wedge\). It links two statements and is similar to the word "and" in English language. A conjunction \(p \wedge q\) requires both parts to be true for the whole expression to be true.
Imagine a scenario where you're evaluating two conditions, \(p\) and \(q\). In this case:

• If both \(p\) and \(q\) are true, then the conjunction \(p \wedge q\) is true.
• If either \(p\) or \(q\) is false, or both are false, then \(p \wedge q\) is false.

The conjunction is a straightforward way to ensure both conditions occur simultaneously for an outcome to be validated as true. In the given exercise, you encounter \(p \wedge \sim q\), where \(\sim q\) means "not \(q\)". This love of detail is vital for evaluating compound statements in logical reasoning.

Truth Values

Truth values are the backbone of logical expressions, dictating whether a statement is valid (true) or not (false). Any logical statement can be boiled down to these two possibilities.
Consider this, in logical expressions such as \(p \rightarrow (p \wedge \sim q)\), assigning truth values to \(p\) and \(q\) allows us to determine the truth of the overall expression:

• "True" (T) indicates that the statement holds.
• "False" (F) suggests the statement does not hold.

In the original exercise, the statements involve expressions where critical attention to truth values determines if complex expressions like implications and conjunctions are true or false. Understanding how truth values drive the whole process helps in precisely verifying logical statements and their outcomes, fostering deeper insights into logical consistency.