Problem 44 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The statement \((p \rightarrow(q \rightarrow p)) \rightarrow(p \rightarrow(p \vee q))\) is: (a) equivalent to \((p \wedge q) \vee(\sim q)\) (b) a contradiction (c) equivalent to \((p \vee q) \wedge(\sim p)\) (d) a tautology

Step-by-Step Solution

Answer

The statement is a tautology.

Step 1: Understanding the Statement Structure

The statement given is \((p \rightarrow(q \rightarrow p)) \rightarrow(p \rightarrow(p \vee q))\). We need to first understand its logical structure. There are two major parts connected by an implication, \(\rightarrow\). The left part is \((p \rightarrow (q \rightarrow p))\) and the right part is \((p \rightarrow (p \vee q))\).

Step 2: Simplifying Each Component

Let's simplify each component. For the left part of the main implication: 1. \((q \rightarrow p)\) is equivalent to \((\sim q \vee p)\) based on logical equivalence of implications. 2. Substitute it into \(p \rightarrow (q \rightarrow p)\), which becomes \(p \rightarrow (\sim q \vee p)\). 3. Use the equivalence \((a \rightarrow b) \equiv (\sim a \vee b)\), which makes it \((\sim p \vee (\sim q \vee p))\). 4. By distribution and logical simplification, this is equivalent to \((\sim p \vee \sim q \vee p)\). Since \((\sim p \vee p)\) is always true, this part reduces to \("True"\). The right part \(p \rightarrow (p \vee q)\) becomes \(\sim p \vee (p \vee q)\), which by distribution simplifies to \((\sim p \vee p) \vee q\). Since \((\sim p \vee p)\) is a tautology (and hence \("True"\)), it further simplifies to \("True"\).

Step 3: Analyzing the Overall Statement

Substituting the simplified components back into the original statement gives us:\[(p \rightarrow(q \rightarrow p)) \rightarrow(p \rightarrow(p \vee q)) \equiv True \rightarrow True\] Since an implication \((True \rightarrow True)\) is also \(\text{True}\), the entire statement becomes \("True"\). This implies that the original statement is a tautology.

Key Concepts

Logical Equivalence

Logical equivalence is an important concept in logic, where two statements have the same truth value. In simpler terms, whenever one statement is true, the other is also true, and whenever one is false, the other one is false too.
To determine logical equivalence, we often use logical rules and identities to transform expressions into more simplified forms.
For example, the implication \((q \rightarrow p)\) can be rewritten as \((\sim q \vee p)\), which preserves the same truth value. This transformation is based on the logical equivalence between an implication and a disjunction.
Logical equivalence is crucial in simplifying expressions and proving statements in mathematics and computer science.

• It allows us to recognize and use statements interchangeably when they exhibit the same logical behavior.
• Knowing different forms of equivalence can aid in recognizing tautologies, contradictions, and other logical properties.

Implication

Implication in logic is an expression represented as \(a \rightarrow b\), which is read as "if a then b". It asserts that whenever \(a\) is true, \(b\) will also be true.
However, if \(a\) is false, the implication \(a \rightarrow b\) is considered true regardless of \(b\)'s truth value. This can sometimes be a confusing concept, but understanding its logical structure is crucial.
Implications can be rewritten to assist in analysis and simplification. For instance, \(p \rightarrow (q \rightarrow p)\) simplifies to \(p \rightarrow (\sim q \vee p)\) using the equivalence relationships.Writing implications in different forms helps to clarify their logic, and this often involves transforming them into disjunctions, as seen with \(\sim p \vee (p \vee q)\).

• Recognizing implications helps understand conditional statements and logical deductions.
• They are fundamental in constructing logical proofs and in reasoning processes.

Logical Simplification

Logical simplification involves reducing complex logical statements into simpler or more intuitive forms without changing their truth values. This is achieved through the application of logical identities and equivalences.
For example, in the step-by-step solution, both parts of the statement were simplified using distribution and other logical rules.
Simplification can be visually appealing and mathematically advantageous as it makes arguments or proofs more straightforward and understandable.
One key aspect of simplification is recognizing tautologies and contradictions. A tautology is a statement that is always true, like \((\sim p \vee p)\). Such components can often be negated or simplified out of the expression.
Simplification frequently involves the use of identities like:

• \(a \vee \sim a \equiv \text{True}\) (the law of excluded middle)
• \(a \wedge \sim a \equiv \text{False}\)

This process not only simplifies expressions but also assists in deeper analysis of the behavior of logical statements and systems.