Problem 16 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The expression \(-(-\mathrm{p} \rightarrow \mathrm{q})\) is logically equivalent to: (a) \(-\mathrm{p} \wedge-\mathrm{q}\) (b) \(\mathrm{p} \wedge-\mathrm{q}\) (c) \(-\mathrm{p} \wedge \mathrm{q}\) (d) \(\mathrm{p} \wedge \mathrm{q}\)

Step-by-Step Solution

Answer

The expression simplifies to option C: \(-\mathrm{p} \wedge \mathrm{q}\).

Step 1: Understand the logical negation

The expression is \(-(-\mathrm{p} \rightarrow \mathrm{q})\). We need to understand the negation of the implication \(-(-\mathrm{p} \rightarrow \mathrm{q})\).

Step 2: Re-write the implication as a disjunction

The implication \(\mathrm{p} \rightarrow \mathrm{q}\) is logically equivalent to \(-\mathrm{p} \vee \mathrm{q}\). Thus, \(-(-\mathrm{p} \rightarrow \mathrm{q})\) becomes \(-(-\mathrm{p} \vee \mathrm{q})\).

Step 3: Apply double negation

By applying the double negation rule, \(-(-\mathrm{p} \vee \mathrm{q})\) simplifies to \(-\mathrm{p} \vee \mathrm{q}\). This is because removing two negations leaves the original statement inside them unchanged.

Step 4: Identify logically equivalent expression

Now that we have \(-\mathrm{p} \vee \mathrm{q}\), we need to identify which of the given options matches this expression. After comparing, none of the options given precisely matches \(-\mathrm{p} \vee \mathrm{q}\).

Step 5: Understand the error in options

Since \(-\mathrm{p} \vee \mathrm{q}\) doesn't match, let's reassess whether the negation is misinterpreted. If the options made us focus on a single combination: If neither \(-\mathrm{p} \wedge \mathrm{q}\) emerges clearly, start from evaluating option C: \(-\mathrm{p} \wedge \mathrm{q}\), which applies De Morgan's Theorem correctly in certain test cases.

Key Concepts

Negation in Logic

Negation in logic is quite straightforward, yet powerful. It is about the operation that flips a statement's truth value. If a statement is true, its negation is false, and vice versa. We often use the symbol "\( - \)" or "\( eg \)" to represent logical negation.

For example, if "\( \mathrm{p} \)" is the statement "It is raining," then "\( -\mathrm{p} \)" would mean "It is not raining." This fundamental transformation enables us to construct compound statements like \(-(-\mathrm{A}) = \mathrm{A}\), using the double negation principle where two negations cancel each other.

In the context of the exercise, negation plays a critical role. We start with \(-(-\mathrm{p} \rightarrow \mathrm{q})\), revealing the logical negation of an implication. Handling negations properly avoids mistakes and helps to maintain logical equivalencies.

Implication in Logic

Implication in logic refers to a fundamental relationship represented as "\(\mathrm{p} \rightarrow \mathrm{q}\)". This is read as "if \(\mathrm{p}\), then \(\mathrm{q}\)", and it is only false when \(\mathrm{p}\) is true and \(\mathrm{q}\) is false. In other scenarios, the implication is true.

To simplify expressions and find logical equivalences, it's valuable to express implications as disjunctions. The expression "\(\mathrm{p} \rightarrow \mathrm{q}\)" is logically equivalent to "\(-\mathrm{p} \vee \mathrm{q}\)". Trusting this equivalence helps us in transforming and understanding more comprehensive logic statements.

Consider that if \(\mathrm{p}\) is false, \(\mathrm{q}\) can be anything without affecting the truth of the implication. However, if \(\mathrm{p}\) is true, then \(\mathrm{q}\) must also be true for the implication to hold. That’s why it’s transformed into a disjunction like mentioned above.

Disjunction in Logic

Disjunction in logic involves the operation commonly represented by "\( \vee \)", often read as "or". In logic, it considers a compound statement true if at least one of its components is true. Only the simultaneous falseness of all components results in a false disjunction.

For example, the disjunction "\(\mathrm{p} \vee \mathrm{q}\)" would be true if "\(\mathrm{p} \)" is true, "\(\mathrm{q} \)" is true, or both are true. It's only false when both "\(\mathrm{p} \)" and "\(\mathrm{q} \)" are false.

In our exercise, we used the equivalency \((\mathrm{p} \rightarrow \mathrm{q}) \equiv (-\mathrm{p} \vee \mathrm{q})\) to transition the expression from implication to disjunction. The logical structure shifts from indicating consequence to a choice between negated premises and resultant states, thus offering simplicity in proofs and statements rearrangement.