Problem 25 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The following statement \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\) is : \(\begin{array}{ll}\text { (a) a fallacy } & \text { (b) a tautology }\end{array}\) (c) equivalent to \(\sim \mathrm{p} \rightarrow \mathrm{q}\) (d) equivalent to \(\mathrm{p} \rightarrow \sim \mathrm{q}\)

Step-by-Step Solution

Answer

The statement is a tautology.

Step 1: Analyze the Statement Structure

The given expression is \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\). This is a complex logical expression made up of implications and negations. We'll check whether it's a tautology, a fallacy, or equivalent to another expression. Our first task is to break this down step by step.

Step 2: Consider Truth Tables

To determine whether the expression is a tautology or fallacy, a truth table is helpful. A tautology is true for all possible truth values of its components, while a fallacy is false for all truth values. Otherwise, it may be logically equivalent to one of the given options, \(~p \rightarrow q\) or \(p \rightarrow ~q\).

Step 3: Construct the Truth Table

Begin by setting up the columns for \(p\), \(q\), and the expressions \(p \rightarrow q\), \(\sim p\), \(\sim p \rightarrow q\), and the main expression \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\). Fill in the truth values for each row based on the logical operators.

Step 4: Compute Individual Components

Evaluate \(p \rightarrow q\) and \(\sim p\) in each row, then use these to compute \(\sim p \rightarrow q\). This produces the columns:- \(p \rightarrow q\): False if \(p\) is true and \(q\) is false, true otherwise.- \(\sim p\): True if \(p\) is false, false otherwise.- \(\sim p \rightarrow q\): False if \(\sim p\) is true and \(q\) is false, true otherwise.

Step 5: Compute Nested Implications

Now evaluate \((\sim p \rightarrow q) \rightarrow q\). This is false only if \(\sim p \rightarrow q\) is true and \(q\) is false, true otherwise. Use results from Step 4 for calculation.

Step 6: Final Evaluation

Evaluate the whole expression \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\) using the previous columns. This final implication is false only if the antecedent \((p \rightarrow q)\) is true and the consequent \([(\sim p \rightarrow q) \rightarrow q]\) is false. Examine these conditions across all rows.

Step 7: Conclusion from Truth Table

By observing the truth table, if the main expression column turns out true in all scenarios, it's a tautology. If false in all, it's a fallacy. If the truth values match those of \(\sim p \rightarrow q\) or \(p \rightarrow \sim q\), it's equivalent to that expression, respectively.

Key Concepts

Tautology

In mathematical logic, a tautology is a statement that is always true, regardless of the truth values of its individual components. Tautologies are like universal truths in logic. They remain constantly valid regardless of how you alter the variables within them.

To understand if a statement is a tautology, you would often use techniques like truth tables. The goal is to examine the statement under every possible combination of truth values for its variables. If it always evaluates to true, then you've identified a tautology!

For example, in the exercise provided, the statement \[(p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\] could be checked for tautology by ensuring every row in its truth table evaluates to true. It's like a way of proving that no matter what the initial conditions, the statement holds strong.

Fallacy

A fallacy in logic is the opposite of a tautology. Instead of always being true, a fallacy is never true. These are statements that, no matter how you play with the truth values of their components, end up false.

When analyzing whether a statement is a fallacy, constructing a truth table comes in handy. By filling in all possible truth combinations, if you find that the statement yields false for every line, then it's identified as a fallacy.

In the exercise statement, to check if it's a fallacy, you would look at each possible scenario of truth values. If the whole statement never turns true, it’s labeled a fallacy. While tautologies always tell a "true story," fallacies always tell "false ones."

Recognizing fallacies helps in verifying the soundness of arguments, ensuring that they do not rely on flawed logic.

Truth Table

A truth table is a mathematical table used to determine the outcome of logical expressions based on their variables' truth values. It's a systematic way to explore all possible scenarios of a logical statement.

Each line in a truth table represents a unique combination of truth values (true or false) for all involved variables. The goal is to fill in the table with the results of the logical operations for these values. This straightforward approach helps in evaluating whether a logical statement is a tautology, fallacy, or something else.

For example, the provided exercise involves constructing a truth table to evaluate the statement \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\). By listing each possible combination of true and false values for \(p\) and \(q\), you can see how the entire expression turns out true or false depending on these values. Truth tables are essential tools in logical reasoning, providing clear insights into the behavior of complex expressions.

Logical Equivalence

Logical equivalence is a fundamental concept in logical reasoning, showing when two statements are exactly the same in terms of their truth values across all possible scenarios. Two expressions are logically equivalent if they always produce the same truth value for any configuration of truth values of their variables.

To determine logical equivalence, you can use a truth table. By comparing the final truth columns of two expressions, if they match in all circumstances, the expressions are logically equivalent.

In the original problem, after examining the truth table, the task is to check if \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\) is equivalent to either \(\sim p \rightarrow q\) or \(p \rightarrow \sim q\).

Understanding logical equivalence is vital, as it allows for transforming expressions into simpler or more useful forms while retaining their inherent meaning. This concept is widely applied in proofs and logical simplifications.