Problem 37 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p\) and \(q\) be two Statements. Amongst the following, the Statement that is equivalent to \(p \rightarrow q\) is |Online May \(\mathbf{1 9 ,} \mathbf{2 0 1 2}\) ] \(\begin{array}{llll}\text { (a) } p \wedge \sim q & \text { (b) } \sim p \vee q & \text { (c) } \sim p \wedge q & \text { (d) } p \vee \sim q\end{array}\)

Step-by-Step Solution

Answer

The statement equivalent to \( p \rightarrow q \) is \( \sim p \vee q \), which is option (b).

Step 1: Understand Implication

The statement \( p \rightarrow q \) is a logical implication which means "if \( p \), then \( q \)." This implication is false only when \( p \) is true and \( q \) is false, and true in all other cases.

Step 2: Use Logical Equivalence

The implication \( p \rightarrow q \) is logically equivalent to \( \sim p \vee q \). This equivalence can be derived from the material implication rule in propositional logic. Essentially, this means that \( p \rightarrow q \) holds true when either \( p \) is false or \( q \) is true.

Step 3: Match with Given Options

Among the given statements, we need to find which one is equivalent to \( \sim p \vee q \). This means looking for the statement representing either \( p \) being false or \( q \) being true.

Step 4: Identify Correct Option

(b) \( \sim p \vee q \) matches exactly with the equivalence derived from \( p \rightarrow q \). Thus, the statement that is equivalent to \( p \rightarrow q \) is option (b).

Key Concepts

Propositional Logic

Propositional logic is a branch of logic that deals with propositions, which are statements that can be either true or false. These propositions are combined through logical connectives to form complex expressions, which can also take on a truth value of true or false. The basic operations within propositional logic include conjunction (\(\wedge\)), disjunction (\(\vee\)), and negation (\(\sim\)).

In our exercise, we deal with two propositions, \(p\) and \(q\). Each can be true or false, and we interpret complex statements that use these basic propositions.

• A conjunction (\(p \wedge q\)) is true only when both \(p\) and \(q\) are true.
• A disjunction (\(p \vee q\)) is true when at least one of \(p\) or \(q\) is true.
• The negation (\(\sim p\)) simply inverts the truth value of \(p\).

Understanding these basics helps us create, interpret, and manipulate more advanced logical statements like implications.

Logical Equivalence

Logical equivalence is a concept where two statements are considered equivalent if they have the same truth value in every possible scenario. In propositional logic, we can transform statements without changing their meaning by using logical equivalences.

For example, the main insight from our exercise is that the statement \(p \rightarrow q\) is logically equivalent to \(\sim p \vee q\). This means for any truth values of \(p\) and \(q\), both expressions will have the same truth value.

Logical equivalence is incredibly powerful because it allows us to replace complex expressions with simpler ones. By doing so, we can solve logical problems more easily and plan logical arguments efficiently.

Material Implication

Material implication in logic refers to the conditional statement \(p \rightarrow q\), which reads as "if \(p\), then \(q\)." This construct captures how truth flows from one proposition to another. Importantly, this statement is considered false only in the case where \(p\) is true, but \(q\) is false; otherwise, it is true.

Understanding this helps make sense of why \(p \rightarrow q\) is equivalent to \(\sim p \vee q\). This equivalence transformation essentially says that the implication holds if either \(p\) is false (thereby not fulfilling the initial promise needed for \(q\) to occur) or \(q\) is true (fulfilling the promise). Therefore, using material implication, we can express logic in more flexible and varied ways while maintaining their meanings.

Truth Tables

Truth tables are critical tools in propositional logic because they allow us to compute the truth value of complex propositions systematically. They display how the truth values of component propositions affect the overall truth value of a logical expression.

To understand \(p \rightarrow q\), a truth table can show its behavior for all possible truth values of \(p\) and \(q\). For instance:

• If \(p\) is true and \(q\) is true, \(p \rightarrow q\) is true.
• If \(p\) is true and \(q\) is false, \(p \rightarrow q\) is false.
• If \(p\) is false and \(q\) is true, \(p \rightarrow q\) is true.
• If \(p\) is false and \(q\) is false, \(p \rightarrow q\) is true.

Using this truth table allows students to visualize and confirm the equivalence between \(p \rightarrow q\) and \(\sim p \vee q\) by comparing each row's result, elucidating how logical operations work.