Problem 11 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The Boolean expression \(\sim(p \Rightarrow(\sim q))\) is equivalent to : (a) \(p \wedge \bar{q}\) (b) \(q \Rightarrow \sim p\) (c) \(p \vee q\) (d) \((\sim p) \Rightarrow q\)

Step-by-Step Solution

Answer

No provided options match the simplified expression \(p \wedge q\).

Step 1: Understand the Given Expression

We start with the given expression \(\sim(p \Rightarrow(\sim q))\). The symbol \(\sim\) means NOT (negation), and \(\Rightarrow\) represents implication.

Step 2: Rewrite Implication Using Logical Equivalences

Recall that any implication \(p \Rightarrow q\) is logically equivalent to \(\sim p \vee q\). Applying this to \(p \Rightarrow(\sim q)\), we have \(\sim p \vee (\sim q)\).

Step 3: Apply Negation to the Expression

Now, negate the expression from Step 2: \(\sim(\sim p \vee \sim q)\). There is a de Morgan's law for negation, which states \(\sim(A \vee B) \equiv \sim A \wedge \sim B\). Apply it: \(\sim(\sim p) \wedge \sim(\sim q)\).

Step 4: Simplify the Double Negation

The double negatives can be simplified because \(\sim(\sim p)\) becomes \(p\) and \(\sim(\sim q)\) becomes \(q\). Thus, \(p \wedge q\) is the simplified expression.

Step 5: Check Against Given Options

Compare the simplified expression \(p \wedge q\) with the given options: (a) \(p \wedge \bar{q}\), (b) \(q \Rightarrow \sim p\), (c) \(p \vee q\), (d) \((\sim p) \Rightarrow q\). None of these options directly match our result, suggesting a misinterpretation. Reconsider previous steps if necessary.

Key Concepts

Logical Equivalences

Logical equivalences are essential tools in Boolean algebra. They are rules that allow us to transform one logical expression into another, simplifying complex logic. In the context of the algebra, these transformations are guaranteed to be true in all scenarios, similar to mathematical tautologies, thereby preserving logical consistency.

• One common equivalence is the relation between implication and disjunction. The implication, denoted as \(p \Rightarrow q\), is equivalent to \(\sim p \vee q\). This transformation allows converting the arrow (\(\Rightarrow\)) operation into more foundational operations—NOT and OR.
• Another key equivalence is related to negating implications. By understanding the transformations, we can simplify or solve logical expressions more effectively.

Logical equivalences help rewrite logical expressions, making them easier to evaluate or prove. They also serve to connect complex logic operations back to simpler operations, ensuring that solutions based on them are both reliable and understandable.
Understanding these foundational transformations is critical when simplifying expressions like the one in the given exercise: \(\sim(p \Rightarrow(\sim q))\). Here, we utilize the equivalence \(p \Rightarrow q \equiv \sim p \vee q\) to help in changing the expression into a different form.

De Morgan's Laws

De Morgan's laws are pivotal in manipulating and simplifying expressions involving negation. Named after Augustus De Morgan, these laws describe how negations interact with the logical operations AND (represented as \(\wedge\)) and OR (represented as \(\vee\)).

• The first law states \(\sim(A \wedge B) \equiv \sim A \vee \sim B\)
• The second law is \(\sim(A \vee B) \equiv \sim A \wedge \sim B\)

These laws allow us to distribute negations inside parentheses and switch from conjunctive contexts (using AND) to disjunctive ones (using OR) and vice versa.
When negating a compound expression, De Morgan's laws provide a systematic approach to finding its equivalent simpler form. In our case, negating \( (\sim p \vee \sim q) \) using De Morgan's law, we get \( \sim(\sim p \vee \sim q) \equiv p \wedge q\). Thus, they proved indispensable in simplifying the Boolean expression in the exercise.
Understanding De Morgan's laws is essential for mastering logic simplification, making them a staple in any logical transformation toolset.

Negation and Implication

Negation and implication are two fundamental operations in Boolean algebra, key to constructing and interpreting logical propositions.

• **Negation (\(\sim\)** usually means reversing the truth value of a proposition. If \(p\) is true, then \(\sim p\) is false, and vice versa.
• **Implication (\(\Rightarrow\)** is used to express logical consequences. If \(p \Rightarrow q\), it means that if \(p\) is true, then \(q\) must also be true.

Understanding how these operations work and interact is necessary for complex logical expressions. For instance, the expression \(\sim(p \Rightarrow(\sim q))\) couples both negation and implication. By recognizing that \(p \Rightarrow q\) is equivalent to \(\sim p \vee q\), the expression can be transformed into a more manageable form, simplifying subsequent steps.
In the exercise, using logical equivalences and negation at the right time led to transforming the expression into \(p \wedge q\), thus illustrating efficient application of these concepts. Mastery of these two operations is invaluable for clear, concise logical reasoning and problem-solving.