Problem 13 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(p \Rightarrow(q \vee r)\) is false, then the truth values of \(p, q, r\) are respectively: (a) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (d) \(\mathrm{F}, \mathrm{F}, \mathrm{F}\)

Step-by-Step Solution

Answer

The truth values are (b) \( \mathrm{T}, \mathrm{F}, \mathrm{F} \).

Step 1: Understanding the Implication

The implication \( p \Rightarrow (q \vee r) \) can be expressed in terms of logical operations. It is equivalent to \( eg p \vee (q \vee r) \). For this to be false, both \( eg p \) and \( q \vee r \) must be false.

Step 2: Analyzing \( \neg p \)

For \( eg p \) to be false, \( p \) must be true since the negation of true gives false.

Step 3: Analyzing \( q \vee r \)

For \( q \vee r \) to be false, both \( q \) and \( r \) must be false, as the OR operation is true unless all operands are false.

Step 4: Combining Conditions

Combining the conditions from Steps 2 and 3, we get: \( p \) is true, \( q \) is false, \( r \) is false. So the truth values must be \( \mathrm{T}, \mathrm{F}, \mathrm{F} \).

Key Concepts

Logical Operations

In logic, various operations combine simple propositions into complex ones, enabling more comprehensive analysis. The fundamental logical operations include:

• AND (\(\wedge\))
• OR (\(\vee\))
• NOT (\(eg\))

In the OR operation (\(q \vee r\)), the resulting expression is true if at least one of the components is true. However, it becomes false only if both components are false.
Logical operations follow specific rules, known as truth tables, dictating the outputs based on different input combinations. Understanding these operations is vital for correctly interpreting and solving logical propositions.

Implication in Logic

Implication in logic, denoted as \(p \Rightarrow q\), represents a conditional relationship where if "p" is true, then "q" must also be true for the entire expression \(p \Rightarrow q\) to be true. Otherwise, the expression becomes false when "p" is true but "q" is false.
Some key points to remember about logical implication:

• If "p" is false, \(p \Rightarrow q\) is always true, regardless of "q".
• If "p" is true and "q" is false, then \(p \Rightarrow q\) is false.
• When both "p" and "q" are true, \(p \Rightarrow q\) remains true.
• In a false implication, logically, it signals an incorrect assumption leading to the result.

Understanding implications is crucial when analyzing logical statements, particularly in mathematics and computer science.

Truth Values in Logic

In logic, truth values signify the validity of propositions. Typically, these are:

• True (T)
• False (F)

Each logical statement or operation results in a truth value that is determined by the truth values of its components. For instance:- An AND operation (\(p \wedge q\)) is true only if both "p" and "q" are true.- An OR operation (\(p \vee q\)) becomes true if any part of it is true, but false only if all components are false.
In the exercise, if the expression \(p \Rightarrow (q \vee r)\) is false, then both conditions have to collectively falsify it. Hence, comprehending how to evaluate truth values based on component propositions and operations will help solve logical problems systematically.

Discrete Mathematics

Discrete mathematics deals with distinct, separate values, often focused on integers and logical statements as opposed to continuous math. Its primary concern lies in structures not requiring continuity, which often appear in computing and algorithm design.
Central topics in discrete mathematics include:

• Logical operations and truth values
• Set theory and combinatorics
• Graph theory
• Algorithms

In discrete mathematics, understanding basic logic operations helps in broader mathematically defined problems, including set operations, proof construction, and algorithmic thinking. Therefore, mastering these logical components is fundamental for grasping complex concepts in discrete math and related computational fields.