Problem 38 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The logically equivalent preposition of \(p \Leftrightarrow q\) is (a) \((p \Rightarrow q) \wedge(q \Rightarrow p)\) (b) \(p \wedge q\) (c) \((p \wedge q) \vee(q \Rightarrow p)\) (d) \((p \wedge q) \Rightarrow(q \vee p)\)

Step-by-Step Solution

Answer

The answer is (a) \((p \Rightarrow q) \wedge(q \Rightarrow p)\).

Step 1: Understanding Logical Equivalence

To solve the problem, we need to find a proposition that is logically equivalent to \(p \Leftrightarrow q\). This symbol means 'if and only if', indicating that both statements \(p\) and \(q\) have the same truth value.

Step 2: Analyzing the Options

Look at the given answer options and understand what each option represents using logical expressions:(a) \((p \Rightarrow q) \wedge (q \Rightarrow p)\): This means that \(p\) implies \(q\) and \(q\) implies \(p\).(b) \(p \wedge q\): This represents both \(p\) and \(q\) being true.(c) \((p \wedge q) \vee (q \Rightarrow p)\): This means either both \(p\) and \(q\) are true, or \(q\) implies \(p\).(d) \((p \wedge q) \Rightarrow (q \vee p)\): This means if both \(p\) and \(q\) are true, then either \(p\) or \(q\) is true.

Step 3: Evaluating Logical Equivalence

The logical equivalence \(p \Leftrightarrow q\) can be expressed as both \((p \Rightarrow q)\) and \((q \Rightarrow p)\) being true at the same time. This is commonly rewritten as option (a): \((p \Rightarrow q) \wedge (q \Rightarrow p)\).

Step 4: Verifying the Solution

Verify each option:- \(p \Rightarrow q\) means if \(p\) is true, then \(q\) must be true.- \(q \Rightarrow p\) means if \(q\) is true, then \(p\) must be true.- The conjunction \((p \Rightarrow q) \wedge (q \Rightarrow p)\) ensures both conditions hold, which is equivalent to saying both have the same truth value, just as \(p \Leftrightarrow q\) does.

Key Concepts

Propositional Logic

Propositional logic is a branch of logic that deals with statements that can be either true or false. These statements are called propositions. Each proposition in logical analysis is typically represented by a letter such as \(p\), \(q\), or \(r\), and can be combined using logical connectives to form more complex statements.
One of the essential elements of propositional logic is understanding the truth values of these propositions and how they interact with each other through these connectives. The basic logical connectives include:

• **AND (\(\wedge\))**: This connective is true only if both propositions are true.
• **OR (\(\vee\))**: This is true if at least one of the propositions is true.
• **NOT (\(eg\))**: This negates the truth value of a proposition.
• **IMPLIES (\(\Rightarrow\))**: This is true unless the first proposition is true and the second is false.
• **BICONDITIONAL (\(\Leftrightarrow\))**: This is true if both propositions have the same truth value.

Logical equivalence is a key concept studied in propositional logic. It involves recognizing when two propositions have the same truth value in every possible scenario. For example, the expression \(p \Leftrightarrow q\) means both propositions must either be true or false for the statement to be true, representing logical equivalence.

Logical Implication

Logical implication is a fundamental concept in logic represented with the symbol \(\Rightarrow\). It connects two propositions, typically denoted as \(p\) and \(q\), in the form of \(p \Rightarrow q\). This is often read as 'if \(p\), then \(q\)' and is only false if \(p\) is true while \(q\) is false.
In practical terms, logical implication means that whenever \(p\) is true, \(q\) must also be true for \(p \Rightarrow q\) to hold. Otherwise, the implication is considered false if \(p\) holds true and \(q\) does not.
The understanding of logical implication is critical when assessing logical equivalence in biconditional statements. In expressions like \((p \Rightarrow q) \wedge (q \Rightarrow p)\), both conditions must independently imply each other, ensuring that both \(p\) and \(q\) always share the same truth value, just like in a biconditional statement (\(p \Leftrightarrow q\)). This dual implication forms the basis of logical equivalence studies.

Biconditional Statement

A biconditional statement is a compound statement formed by the combination of two conditionals or implications. It is represented by the symbol \(\Leftrightarrow\) and is used to express statements where both component propositions \(p\) and \(q\) have identical truth values.
The meaning of \(p \Leftrightarrow q\) is '\(p\) if and only if \(q\)'. This means that \(p\) implies \(q\) and \(q\) implies \(p\). It is equivalent to saying \((p \Rightarrow q) \wedge (q \Rightarrow p)\), where each implication has to be true for the biconditional to be true.
Thus, the biconditional statement is true in two scenarios:

• Both \(p\) and \(q\) are true.
• Both \(p\) and \(q\) are false.

If one has a different truth value than the other, the biconditional statement becomes false. This understanding helps in various logical analyses and is integral when verifying logical equivalence, as shown in the step-by-step solution where option \(a\) represents it correctly as \((p \Rightarrow q) \wedge (q \Rightarrow p)\).