Mathematical Reasoning

Statement: a sentence that is either true or false, but not both. (Exclamations, questions, opinions don't qualify.)

Examples:

• "2 + 2 = 4" ✓ (true statement)
• "The square root of 9 is 4" ✓ (false statement)
• "Open the door!" ✗ (not a statement — imperative)
• "Is it raining?" ✗ (question)
• "x + 5 = 8" — depends on x; not a statement until x is specified.

LOGICAL CONNECTIVES

Let p, q be statements.

1. Negation: ~p (or ¬p) — opposite of p.

• p: "It is raining" → ~p: "It is not raining."

2. Conjunction: p ∧ q ("p AND q") — true iff both are true.

3. Disjunction: p ∨ q ("p OR q") — inclusive or. True if at least one is true.

4. Implication / Conditional: p → q ("if p, then q").

• False only when p is true and q is false. Else true.
• p is the antecedent (hypothesis); q is the consequent.

5. Biconditional: p ↔ q ("p if and only if q"). True when both have the same truth value.

Truth tables:

Note: F → anything is true. (Vacuously true.) "If 2 + 2 = 5, then I am the king of France" — technically true.

Related forms of an implication p → q:

• Converse: q → p.
• Inverse: ~p → ~q.
• Contrapositive: ~q → ~p.

Important equivalences:

• An implication and its contrapositive are LOGICALLY EQUIVALENT.
  Example: "If it's a square, it's a rectangle" ⟺ "If it's not a rectangle, it's not a square."
• An implication and its converse are NOT equivalent.

TAUTOLOGY, CONTRADICTION, CONTINGENCY

• Tautology: always true regardless of truth values. Example: p ∨ ~p.
• Contradiction: always false. Example: p ∧ ~p.
• Contingency: sometimes true, sometimes false.

Some standard tautologies:

• p → p
• p ∨ ~p (law of excluded middle)
• ~(p ∧ ~p) (law of non-contradiction)
• (p ∧ (p → q)) → q (modus ponens form)
• ((p → q) ∧ ~q) → ~p (modus tollens form)
• (p → q) ↔ (~q → ~p) (contrapositive)
• (p → q) ↔ (~p ∨ q)
• (p ∧ q) ↔ (p ∨ ~q) (De Morgan)
• (p ∨ q) ↔ (p ∧ ~q) (De Morgan)

QUANTIFIERS

• Universal: ∀ ("for all"). ∀x P(x) means P(x) holds for every x.
• Existential: ∃ ("there exists"). ∃x P(x) means at least one x satisfies P.

Negation of quantifiers:

• (∀x P(x)) ≡ ∃x (P(x)) (negate the universal → existential negation)
• (∃x P(x)) ≡ ∀x (P(x))

Example: "Every student passed" — negation: "There exists a student who did NOT pass."

METHODS OF PROOF

1. Direct proof: assume p, derive q.
2. Contrapositive: prove ~q → ~p (equivalent to p → q).
3. Contradiction: assume p and ~q together; derive contradiction.
4. Counterexample: to disprove ∀x P(x), exhibit one x where P fails.
5. Mathematical induction: for statements about natural numbers (Pack 8).

Worked example. Prove by contradiction: there are infinitely many primes (Euclid).

Suppose finitely many primes: p₁, p₂, ..., p_n. Form N = p₁p₂...p_n + 1. N is not divisible by any p_i (gives remainder 1). So N has a prime factor different from p₁, ..., p_n — contradicting "finitely many." Hence infinitely many primes. QED.

JEE-style questions typically ask:

• Find the contrapositive / converse / inverse of a given statement.
• Determine if a compound statement is tautology / contradiction.
• Negation of quantified statement.