Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

Truth value of the statement "if \(p\) then \(q\) " is false when (A) \(p\) is true, \(q\) is true (B) \(p\) is true, \(q\) is false (C) \(p\) is false, \(q\) is true (D) \(p\) is false, \(q\) is false

Step-by-Step Solution

Answer

The statement is false when (B) \(p\) is true, \(q\) is false.

Step 1: Understanding Conditional Statements

A conditional statement "if \(p\), then \(q\)" is often denoted as \(p \rightarrow q\). The statement is true in most cases except when \(p\) is true and \(q\) is false.

Step 2: Evaluating Each Option

Evaluate each option to determine the truth value of the statement \(p \rightarrow q\):- (A) \(p\) is true, \(q\) is true: The statement is true.- (B) \(p\) is true, \(q\) is false: The statement is false.- (C) \(p\) is false, \(q\) is true: The statement is true.- (D) \(p\) is false, \(q\) is false: The statement is true.

Step 3: Identifying the False Condition

From the evaluations, option (B) where \(p\) is true and \(q\) is false, is the condition under which the conditional statement \(p \rightarrow q\) is false.

Key Concepts

Conditional Statements

Conditional statements are fundamental in mathematical logic and everyday reasoning. They describe a relationship between two statements, where one statement is said to lead to another. You might know conditional statements more familiarly as "if-then" statements. For example, "if it rains, then the ground will be wet."
A basic conditional statement has the form "if \(p\), then \(q\)" written symbolically as \(p \rightarrow q\). Here, \(p\) is known as the hypothesis or antecedent, while \(q\) is the conclusion or consequent.

• If \(p\) is true and \(q\) is true, then the statement \(p \rightarrow q\) is true.
• If \(p\) is true and \(q\) is false, the statement is false. This is because the expected conclusion \(q\) did not happen after \(p\) occurred.
• If \(p\) is false, the statement \(p \rightarrow q\) is true, regardless of \(q\)'s truth value. This might seem counterintuitive but is because nothing is promised if the condition \(p\) isn't met.

Conditional statements are powerful as they allow us to generalize and make logical deductions.

Logic in Mathematics

Logic in mathematics is the formal scientific study of the principles of valid inference and reasoning. It concerns itself with propositions, which are statements that can be either true or false. Logical reasoning is the backbone of mathematical proofs and problem solving.
Mathematical logic uses different types of statements and operators to form new statements from existing ones. These include:

• Conjunction (AND): A statement "\(p \land q\)" is true if both \(p\) and \(q\) are true.
• Disjunction (OR): "\(p \lor q\)" is true if at least one of \(p\) or \(q\) is true.
• Negation (NOT): The statement "not \(p\)" is true when \(p\) is false.
• Conditional (IMPLIES): "\(p \rightarrow q\)" is true except when \(p\) is true and \(q\) is false.

Understanding these basic operators and how they interact is crucial for developing strong logical and deductive skills in mathematics. Logical thinking helps us to structure our reasoning clearly.

Propositional Logic

Propositional logic, also known as statement logic or sentential logic, is the branch of logic that deals with the study and analysis of propositions and their connections to each other via logical connectives.
In propositional logic, complex statements are formed using simple propositions connected by logical operators like AND (\(\land\)), OR (\(\lor\)), NOT (\(eg\)), and IMPLIES (\(\rightarrow\)). These operators help in constructing compound statements, which can represent complex logical relationships.

• Each simple proposition holds a truth value: true or false.
• Connectives allow for combining propositions to express more nuanced true or false statements.
• Propositional logic does not account for the internal structure of propositions, focusing instead on the logical flow.

This form of logic is instrumental in computer science, mathematics, and philosophy for developing algorithms, proving theorems, and checking the consistency of systems. It lays the groundwork for more advanced topics in logic and computer science.