Problem 49 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The negation of the Boolean expression \(\sim s \vee(\sim r \wedge s)\) is equivalent to: \(\quad\) (a) \(\sim s \wedge \sim r\) (b) \(r\) (c) \(s \vee r\) (d) \(s \wedge r\)

Step-by-Step Solution

Answer

The negation is equivalent to \( s \wedge r \).

Step 1: Understand the Original Statement

We are given the Boolean expression \( \sim s \vee (\sim r \wedge s) \). To negate this, we need to apply De Morgan's laws and other logical equivalences.

Step 2: Apply the Laws of Negation

To negate \( \sim s \vee (\sim r \wedge s) \), use De Morgan's Laws:\[\sim (A \vee B) = \sim A \wedge \sim B\]Here, \(A = \sim s\) and \(B = (\sim r \wedge s)\). Therefore, the negation becomes:\[\sim (\sim s) \wedge \sim (\sim r \wedge s)\]

Step 3: Simplify the Negations

Simplify \(\sim (\sim s)\) to \(s\), and apply De Morgan's law to \(\sim (\sim r \wedge s)\):\[\sim (\sim r \wedge s) = \sim (\sim r) \vee \sim s = r \vee \sim s\]Thus, the expression becomes:\[s \wedge (r \vee \sim s)\]

Step 4: Apply Distribution

Apply the distribution of \(s\) over \( (r \vee \sim s) \) using the distributive law:\[s \wedge (r \vee \sim s) = (s \wedge r) \vee (s \wedge \sim s)\]Since \(s \wedge \sim s = \text{false}\), it simplifies to:\[s \wedge r\]

Step 5: Conclusion

The negation of the original expression \( \sim s \vee (\sim r \wedge s) \) simplifies to \( s \wedge r \). Therefore, the correct answer is (d) \( s \wedge r \).

Key Concepts

De Morgan's Laws

De Morgan's Laws are fundamental principles in Boolean algebra that help in transforming logic expressions. These laws provide a way to negate the conjunction (AND) and disjunction (OR) operations. Here's a simpler way to look at them:

• First Law: The negation of a conjunction. This is written as: \( \sim (A \wedge B) = \sim A \vee \sim B \). It shows that negating an AND operation results in an OR operation with each component negated.
• Second Law: The negation of a disjunction. Expressed as: \( \sim (A \vee B) = \sim A \wedge \sim B \). This means negating an OR operation results in an AND operation with each part negated.

By applying these laws, any complex logic expression can be simplified or transformed, making it easier to work with. In the original exercise, these laws helped negate the expression \( \sim s \vee (\sim r \wedge s) \) effectively.
To apply De Morgan's laws on this expression, you first recognize it as a whole (like \( A \vee B \)), and apply negation to each part, switching the operation from OR to AND.

Negation in Logic

Negation in logic is a crucial operation that inverts the truth value of a given proposition. If the statement is true, its negation will be false, and vice versa. This operation is commonly symbolized by \( \sim \) or "not."

• Basic Concept: Think of negation as a logical mirror. Whatever truth value you start with is flipped. For example, \( \sim \text{True} = \text{False} \) and \( \sim \text{False} = \text{True} \).
• Negating Combined Expressions: When dealing with combined expressions (like those with AND or OR operators), negation affects not only the individual elements but also the operation connecting them. De Morgan's laws explain how the operators switch: AND becomes OR, and OR becomes AND, when negated.

In the exercise, negation was applied to the expression \( \sim s \vee (\sim r \wedge s) \). This involved negating each part, understanding the logical connections, and simplifying the entire expression step-by-step to reach the conclusion.

Distributive Law in Logic

The distributive law in logic is analogous to the distributive property in arithmetic. It shows how logical operations are rearranged, specifically demonstrating how an AND can distribute over OR or vice versa.

• Distributive Formula for Logic: \[ A \wedge (B \vee C) = (A \wedge B) \vee (A \wedge C) \]
• Alternate Case: \[ A \vee (B \wedge C) = (A \vee B) \wedge (A \vee C) \]

When it comes to Boolean expressions, leveraging the distributive law helps in simplifying or refactoring expressions to make them more intuitive or in a desired canonical form.
In the step-by-step solution provided, this law was employed to distribute \( s \) over the expression \( r \vee \sim s \). This created two separate terms, simplifying further analysis: \( (s \wedge r) \vee (s \wedge \sim s) \). Although \( s \wedge \sim s \) evaluates to false (since \( s \) cannot be both true and false), this demonstrates the use of the distributive law as a powerful tool in simplifying expressions.