Problem 28 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The negation of \(\sim \mathrm{s} \vee(\sim \mathrm{r} \wedge \mathrm{s})\) is equivalent to: (a) \(s \vee(r \vee \sim s)\) (b) s\wedger (c) \(\mathrm{s} \wedge \sim \mathrm{r}\) (d) \(s \wedge(\mathrm{r} \wedge \sim \mathrm{s})\)

Step-by-Step Solution

Answer

The negation is equivalent to option (d): \(s \wedge (r \wedge \sim s)\).

Step 1: Understand the Given Expression

The given expression is \( \sim s \vee (\sim r \wedge s) \). We need to find its negation.

Step 2: Apply De Morgan's Laws

According to De Morgan's Laws, the negation of a disjunction \( eg (A \vee B) \) is equivalent to \( eg A \wedge eg B \). Applying this, the negation of \( \sim s \vee (\sim r \wedge s) \) will be \( eg(\sim s) \wedge eg (\sim r \wedge s) \).

Step 3: Simplify Negation of Components

\( eg(\sim s) \) simplifies to \( s \) and \( eg(\sim r \wedge s) \) simplifies using De Morgan's Laws again to \( \sim (\sim r) \vee \sim s \), which is equivalent to \( r \vee \sim s \).

Step 4: Combine the Simplified Components

Combine the components: \( s \wedge (r \vee \sim s) \).

Step 5: Final Equivalence Check

The negation \( s \wedge (r \vee \sim s) \) matches option (d) from the given list.

Key Concepts

De Morgan's Laws

De Morgan's Laws are fundamental in mathematical logic and Boolean algebra. These laws provide a way to simplify and manipulate logical expressions by transforming conjunctions into disjunctions and vice versa. The laws are used when dealing with the negation of complex expressions.

Let's break down the two primary transformations De Morgan's Laws offer:

• The negation of a conjunction: The negation of an expression like \((A \wedge B)\) translates into \(\sim A \vee \sim B\).
• The negation of a disjunction: For a disjunction expression \((A \vee B)\), its negation is \(\sim A \wedge \sim B\).

In our original exercise, De Morgan's Laws help us transform the negation of \(\sim s \vee (\sim r \wedge s)\) by changing the expression structure. This systematic approach allows for clear reasoning and a pathway toward simplifying logical propositions.

Logical Negation

In mathematical logic, negation is a pivotal operation that flips the truth value of a statement. If something is true, negation makes it false, and vice versa.

When dealing with complex logical statements, applying logical negation isn't just about changing 'true' to 'false'. It involves a deeper understanding of how each part of a compound statement interacts:

• If the statement is an atomic one, like \(p\), its negation is simply \(\sim p\).
• When the statement involves operators, such as conjunction or disjunction, De Morgan's Laws assist in properly distributing the negation across the expression.

For instance, in the original solution, \(\sim s\) becomes \(s\) when negated, and the compound expression \(\sim r \wedge s\) requires flipping through De Morgan's Laws: first applying negation to both \(\sim r\) and \(s\), transforming it to \( r \vee \sim s \). Logical negation is essential for understanding how truth values are inverted across more complex logical frameworks.

Boolean Algebra

Boolean algebra is the branch of algebra where the values of the variables are true and false, typically denoted as 1 and 0, respectively.

This type of algebra is foundational for digital circuits and computing because it handles binary values effectively through logical operations like AND (\(\wedge\)), OR (\(\vee\)), and NOT (\(\sim\)).

• AND operations correspond to logical conjunctions, where a true outcome results only if all operands are true.
• OR operations correspond to logical disjunctions, allowing a true output if any operand is true.
• NOT or negation inversely switches the truth value of its operand.

In the given exercise, Boolean algebraic principles are applied to determine the equivalence of multiple logical statements. Through Boolean manipulation, the original statement \(\sim s \vee (\sim r \wedge s)\) was simplified and its negation found, ultimately leading to the equivalent expression in the form \(s \wedge (r \vee \sim s)\). Recognizing and practicing these skills in Boolean algebra helps in designing logic gates and solving logical problems effectively.