Problem 31 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The proposition \(\sim(p \vee-q) \vee \sim(p \vee q)\) is logically equivalent to: (a) \(p\) (b) \(q\) (c) \(\sim p\) (d) \(\sim q\)

Step-by-Step Solution

Answer

The given proposition is logically equivalent to (c) \(\sim p\).

Step 1: Apply De Morgan's Law

Recall De Morgan's Law: \[ \sim(p \vee q) \equiv \sim p \wedge \sim q \]\ Use De Morgan's Law to transform the first negation: \[ \sim(p \vee -q) \] becomes \[ \sim p \wedge \sim(-q) \] which simplifies to \[ \sim p \wedge q \] by noting that \(\sim(-q)\) is \(q\).

Step 2: Use De Morgan's Law Again

Apply De Morgan's Law to the second part: \[ \sim(p \vee q) \] becomes \[ \sim p \wedge \sim q \].

Step 3: Apply Distributive Law

Now consider the expression with negations applied: \[ (\sim p \wedge q) \vee (\sim p \wedge \sim q) \]Use distribution over \( \vee \):Factor out \(\sim p\): \[ \sim p \wedge (q \vee \sim q) \]

Step 4: Simplify Using Tautology

Recognize that the expression \(q \vee \sim q\) is a tautology, meaning it is always true. Therefore:\[ \sim p \wedge (q \vee \sim q) \] simplifies to \(\sim p\).

Step 5: Conclusion

Thus, the original proposition \(\sim(p \vee -q) \vee \sim(p \vee q)\) is logically equivalent to \(\sim p\).

Key Concepts

Propositional Logic

Propositional logic is a branch of logic that deals with propositions, which are statements that can either be true or false. In propositional logic, we utilize variables such as \( p \) and \( q \) to represent different propositions. Logical operations such as AND (\( \wedge \)), OR (\( \vee \)), and NOT (\( \sim \)) are used to form more complex expressions from these propositions.
For example, \( \sim p \) signifies "not \( p \)", indicating that if \( p \) is true, then \( \sim p \) is false, and vice versa. Through the combination of these operations, propositional logic allows us to evaluate the truth values of compound statements.

De Morgan's Laws

De Morgan's Laws are crucial rules in propositional logic that facilitate the process of transforming logical expressions. These laws help simplify complicated propositions, making the logic easier to manage. They state the following:

• \( \sim (p \vee q) \equiv (\sim p \wedge \sim q) \)
• \( \sim (p \wedge q) \equiv (\sim p \vee \sim q) \)

Using De Morgan's Laws, we can transform a negation of an OR operation into an AND operation with negations, and vice versa. In the solution, we applied De Morgan's Laws to transform \( \sim (p \vee -q) \) into \( \sim p \wedge q \), which reshapes the expression, thereby simplifying the logical equation.
This is a fundamental technique enabling us to manipulate and find logical equivalences.

Tautology

In logical terms, a tautology is a statement that holds true under every possible interpretation. No matter how the individual propositions are evaluated, a tautology remains true. A common example of a tautology is the expression \( q \vee \sim q \). This expression will always evaluate to true since at least one of \( q \) or \( \sim q \) must be true at any given time.
In the context of the solution, recognizing \( q \vee \sim q \) as a tautology allowed us to greatly simplify our expression to just \( \sim p \). Understanding tautologies is key in simplifying logical expressions, as they reveal parts of a logic statement that don't influence the overall truth value.

Distributive Law

The distributive law is one of the rules of propositional logic that involves distributing a logical operator across another operation. It's similar to the distributive property in arithmetic, where \( a(b + c) = ab + ac \). In logic, the distributive law can be expressed with AND and OR operations:

• \( p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r) \)
• \( p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r) \)

During the solution, we used the distributive law to factor out \( \sim p \) in the expression \( (\sim p \wedge q) \vee (\sim p \wedge \sim q) \) to achieve \( \sim p \wedge (q \vee \sim q) \). This law helped to simplify the overall logic operation, making it manageable and revealing the essential components of the expression. By distributing and factoring as appropriate, we unveil the logical equivalence hidden in complex expressions.