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Visual SolutionTricky
Question
The negation of the statement 'If I become a teacher, then I will open a school' is:
(A)If I will not open a school, then I will not become a teacher
(B)I will not become a teacher or I will open a school
(C)I will become a teacher and I will not open a school
(D)I will not become a teacher and I will not open a school
Solution Path
Negation of conditional p→qp \to q is p∧∼qp \wedge \sim q (not the contrapositive). Apply: 'I will become a teacher AND I will not open a school'.
01Question Setup
1/4
Find the negation of: 'If I become a teacher, then I will open a school'. Identify this as a conditional p→qp \to q.
Identify pp: become a teacher, qq: open a school
02Negation of Conditional
2/4
The negation of p→qp \to q is NOT ∼p→∼q\sim p \to \sim q (inverse). The correct negation is p∧∼qp \wedge \sim q. Keep the hypothesis and negate only the conclusion.
∼(p→q)=p∧∼q\sim(p \to q) = p \wedge \sim q
03Trap: Contrapositive vs NegationKEY INSIGHT
3/4
The contrapositive ∼q→∼p\sim q \to \sim p is EQUIVALENT to the original (not its negation). Students confuse contrapositive with negation. Negation uses AND, not IF-THEN.
Contrapositive $
eqNegation. Negation. \sim(p \to q) = p \wedge \sim q$
04Final Answer
4/4
Applying ∼(p→q)=p∧∼q\sim(p \to q) = p \wedge \sim q: 'I WILL become a teacher AND I will NOT open a school'.
Answer (C)\boxed{\text{Answer (C)}}: I will become a teacher and I will not open a school
Concepts from this question2 concepts unlocked
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Negation Rules for Compound Statements

EASY

To negate a compound statement, apply De Morgan's laws: negate each component and swap AND with OR (and vice versa). For conditionals, ~(p -> q) = p AND ~q.

∼(p∧q)=∼p∨∼q,∼(p→q)=p∧∼q\sim(p \wedge q) = \sim p \vee \sim q, \quad \sim(p \to q) = p \wedge \sim q

Negation questions appear in almost every JEE Main paper on this topic. Getting De Morgan's wrong costs easy marks. The conditional negation formula is especially tricky since students default to ~p -> ~q.

Negation problemsStatement equivalenceContrapositive identification
Practice (10 Qs) →

Conditional Statement Equivalences

STANDARD

p -> q is logically equivalent to ~p OR q, and also to its contrapositive ~q -> ~p. The conditional is false only when p is true and q is false.

p→q≡∼p∨q≡∼q→∼pp \to q \equiv \sim p \vee q \equiv \sim q \to \sim p

Converts conditional problems into simpler OR statements. The equivalence p -> q = ~p OR q is the single most tested identity in this chapter.

Truth value problemsLogical equivalenceTautology identification
Practice (9 Qs) →
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