Problem 30 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(p, q, r\) denote arbitrary statements. Then the logically equivalent of the statement \(p \Rightarrow(q \vee r)\) is: (a) \((p \vee q) \Rightarrow \mathrm{r}\) (b) \((p \Rightarrow q) \vee(p \Rightarrow r)\) (c) \((p \Rightarrow \sim q) \wedge(p \Rightarrow r)\) (d) \((p \Rightarrow q) \wedge(p \Rightarrow \sim r)\)

Step-by-Step Solution

Answer

The logically equivalent statement is (b): \((p \Rightarrow q) \vee(p \Rightarrow r)\).

Step 1: Understand the Implication

The given statement is an implication: \( p \Rightarrow (q \vee r) \). This means if \( p \) is true, then \( q \vee r \) must also be true.

Step 2: Apply Logical Equivalence of Implication

Recall that the statement \( A \Rightarrow B \) is logically equivalent to \( \sim A \vee B \). Thus, \( p \Rightarrow (q \vee r) \) can be rewritten as \( \sim p \vee (q \vee r) \).

Step 3: Simplify Using Distributive Law

Use the associative and distributive laws to simplify: \( \sim p \vee (q \vee r) \equiv (\sim p \vee q \vee r) \).

Step 4: Identify Equivalent Statement

From the rewritten form \( \sim p \vee q \vee r \), look for an answer choice that matches. The logical equivalence of distributive and De Morgan's laws suggests that this is equivalent to statement (b): \((p \Rightarrow q) \vee(p \Rightarrow r)\).

Step 5: Verify with Truth Table

For full verification, you could construct a truth table for both \( p \Rightarrow (q \vee r) \) and \((p \Rightarrow q) \vee(p \Rightarrow r)\), finding that they're always the same under all combinations of truth values for \( p, q, \) and \( r \).

Key Concepts

Logical Implication

In logic, an implication, like the statement \( p \Rightarrow (q \vee r) \), is a fundamental construct that needs clear understanding. An implication is a type of conditional statement. It can be interpreted as "if \( p \) is true, then \( q \vee r \) must also be true." This means that whenever \( p \) happens, at least one of \( q \) or \( r \) should occur, or both.

Logically, an implication \( A \Rightarrow B \) can be rewritten using logical equivalence to \( \sim A \vee B \). Here, \( \sim A \) indicates "not A," making the implication equivalent to saying "either A is false, or B is true." This transformation allows us to visualize implications in terms of logical OR operations, facilitating easiness in manipulations and simplifications of logical expressions.

Truth Table

A truth table is a helpful tool in logic to evaluate the truth values of propositions and their combinations. It systematically lists all possible combinations of truth values for the involved statements and shows what each expression evaluates to under those conditions.

For our implication \( p \Rightarrow (q \vee r) \), a truth table would consider all possible truth values of \( p, q, \) and \( r \). We compare these values to those produced by \((p \Rightarrow q) \vee (p \Rightarrow r)\) to confirm they result in identical outcomes.

Truth tables, though conceptually straightforward, are powerful in verifying logical equivalences, such as determining if two expressions yield the same results across all combinations of input values.

De Morgan's Laws

De Morgan's Laws are crucial in understanding how to distribute negations over logical operators. These laws provide a way to transform complex logical statements into simpler equivalent forms. They are stated as:

• \( \sim (A \wedge B) \equiv (\sim A \vee \sim B) \)
• \( \sim (A \vee B) \equiv (\sim A \wedge \sim B) \)

Applying De Morgan's Laws can be especially useful in the step-by-step simplification of logical expressions, though it is not directly used in the solution to our original problem, understanding it is essential as it frequently reoccurs in logical transformations.

Often, they serve in simplifying negations by transforming conjunctions and disjunctions, offering clearer insight into the logical structure of complex expressions.

Distributive Law

The distributive law in logic allows us to distribute one operator over another, often simplifying complex logical expressions. This law is similar to the algebraic distributive property, but it applies to logical operators. Specifically, it tells us:

• \( A \wedge (B \vee C) \equiv (A \wedge B) \vee (A \wedge C) \)
• \( A \vee (B \wedge C) \equiv (A \vee B) \wedge (A \vee C) \)

For the problem at hand, we employed this law to modify the expression \( \sim p \vee (q \vee r) \), though it was a simple form in this context. Understanding the distributive law helps in recognizing how logical operators can interact and combine, making it a valuable tool for working through and simplifying larger logical expressions.