Mathematical Reasoning
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Direct answer
A statement in logic is a sentence that is exactly true or false, never both; compound statements are built with and, or, not and implies. The two equivalences that decide almost every JEE question are: p implies q equals its contrapositive not q implies not p, and the negation of "for all x, p(x)" is "there exists x for which p(x) is false". Truth tables settle everything; standard equivalences settle it faster.
What you must remember
- Compound statements: conjunction (p and q), disjunction (p or q, inclusive by mathematical default), negation (not p); fluency in their truth tables is assumed.
- Implication family: for p implies q, the converse is q implies p, the inverse is not p implies not q, and the contrapositive is not q implies not p; the implication and its contrapositive are equivalent, as are the converse and inverse.
- Negations of compounds: not (p and q) = (not p) or (not q); not (p or q) = (not p) and (not q); not (p implies q) = p and (not q) — the last is the standard trap.
- Quantifiers: not (for all x, p(x)) = there exists x with (not p); not (there exists x, p(x)) = for all x, (not p) — push the negation inward past the quantifier and flip it.
- Tautology and contradiction: p or (not p) is always true, p and (not p) always false; a compound built only from tautologies is itself a tautology.
- Contrapositive method: to prove "if n^2 is even then n is even", prove "if n is odd then n^2 is odd" — equivalent and easier.
- Validity checking: two statements are equivalent when their truth tables agree column by column; one counterexample kills a claimed equivalence.
Common confusion
Negating an implication is the perpetual slip: not (p implies q) is not the inverse — it is p and not q, because an implication fails only when the premise holds yet the conclusion does not. Students also swap the converse and the contrapositive under time pressure; only the contrapositive is equivalent to the original. With quantifiers, for-all must flip to there-exists and vice versa; leaving the quantifier unflipped is a silent error.
Exam-focused takeaway
Mathematical reasoning is a JEE Main chapter: questions ask for the contrapositive or converse of a given statement, the negation of a quantified statement, truth-table classification of a compound as tautology, contradiction or neither, and equivalence checking. Marks are quick but demand literal reading — exactly one option matches the definition, and imprecise language is how marks bleed. Read each option as a truth-table column, not prose.
Frequently asked questions
What is the contrapositive of p implies q?
Not q implies not p — logically equivalent to the original implication; converse and inverse are equivalent to each other, not to the original.
Is the converse of a true implication true?
Not necessarily. "If x = 2 then x^2 = 4" is true while its converse fails at x = -2; equivalence of an implication and its converse must be checked, not assumed.
What is the negation of "for all x, p(x) is true"?
"There exists x for which p(x) is false" — quantifier flips, property negates; one counterexample certifies it.
What is the negation of p implies q?
p and (not q) — the only situation in which the implication fails; the inverse is a different statement, not the negation.
What is a tautology?
A compound statement true under every assignment of its components, such as p or (not p); a contradiction is false under every assignment, such as p and (not p).