Negation — Of compound statements
Mathematical Reasoning
7
JEE Qs
8%
Hard
45
min
Master De Morgan's Laws and the negation of conditional statements, paying close attention to correctly applying rules for quantifiers, to avoid common traps.
🧮 Key Formulas
✅ Key Points for JEE
- 1De Morgan's laws are fundamental for negating 'AND' and 'OR' statements: switch the connective (∧ to ∨, ∨ to ∧) and negate each individual component statement.
- 2The negation of an implication 'P → Q' is 'P ∧ ¬Q'; a common mistake is to write '¬P → ¬Q' which is incorrect.
- 3When negating statements involving quantifiers, 'for all (∀)' becomes 'there exists (∃)' and vice-versa, while the predicate itself is also negated.
- 4Negating complex compound statements often requires multiple steps, applying the negation rules hierarchically from the outermost connective inwards.
⚠️ Common Mistakes
- ✕Incorrectly applying De Morgan's laws by failing to switch the connective (e.g., negating P ∧ Q to ¬P ∧ ¬Q instead of ¬P ∨ ¬Q).
- ✕Mistaking the negation of 'if P then Q' for 'if not P then not Q' (inverse) or 'not P then Q'.
- ✕Failing to negate the quantifier (e.g., keeping 'for all' as 'for all' instead of changing to 'there exists') or incorrectly negating the predicate when quantifiers are involved.
📝 Practice Questions
See allQ68. Choose the correct answer from the options given below : (1) (A)-(IV), (B)-(III), (C)-(II), (D)-(I) (2) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
Q64. Choose the correct answer from the options given below : (1) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (2) (A)-(I), (B)-(III), (C)-(II), (D)-(IV) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
Q63. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below : (1) Both Statement I and Statement II are true (2) Statement I is false but Statement II is true (3) Statement I is true but Statement II is false (4) Both Statement I and Statement II are false
Q73.Negation of (p →q) →(q →p) is (1) (p~) ∨p (2) q ∧(~p) (3) (~q) ∧p (4) p ∨(~q)
Q73.Among the statements (S1) : (p ⇒q) ∨((~p) ∧q) is a tautology (S2) : (q ⇒p) ⇒((~p) ∧q) is a contradiction (1) Neither (S1) and (S2) is True (2) Both (S1) and (S2) are True (3) Only (S2) is True (4) Only (S1) is True
Q72.Which of the following statements is a tautology? (1) p →(p ∧(p →q)) (2) (p ∧q) →(~(p) →q) (3) (p ∧(p →q)) →~q (4) p ∨(p ∧q)
NCERT Chapters
- Class 11 Maths Ch 14: Mathematical Reasoning