Mathematical Reasoning formulas for JEE Main & Advanced

13 formula entries for Mathematical Reasoning, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Conjunction (AND)

p ∧ q is T iff p = T and q = T

When to use. Combine two conditions that must both hold.

Trap. Conjunction is commutative: p ∧ q ≡ q ∧ p.

Disjunction (OR)

p ∨ q is F iff p = F and q = F

When to use. Combine conditions where at least one must be true.

Trap. Inclusive OR: both true gives true.

Negation (NOT)

~p = T iff p = F

When to use. Form the opposite of a statement.

Trap. Negation of a compound statement requires De Morgan's laws.

Conditional (Implication) p → q

p → q ≡ ~p ∨ q

When to use. Express 'if p then q' in logical reasoning.

Trap. Do not confuse with causality; it's logical implication.

Biconditional p ↔ q

p ↔ q ≡ (p → q) ∧ (q → p)

When to use. Express equivalence or necessary and sufficient conditions.

Trap. It is equivalent to 'if and only if'.

Logical Equivalence

p ≡ q means p ↔ q is a tautology

When to use. Simplify or replace statements with equivalent forms.

Trap. Equivalence is not the same as implication.

De Morgan's Laws

~(p ∧ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∧ ~q

When to use. Negate compound statements with AND/OR.

Trap. Do not forget to negate both components.

Converse, Inverse, Contrapositive

p → q ≡ ~q → ~p

When to use. Rewrite implications for proofs or logical reasoning.

Trap. Converse and inverse are not equivalent to original.

Quantifiers: Universal (∀) and Existential (∃)

∀x P(x): true if P(x) holds for every x in domain. ∃x P(x): true if P(x) holds for at least one x.

When to use. Express statements about all or some elements in a set.

Trap. Negation of ∀ is ∃ with negation: ~(∀x P(x)) ≡ ∃x ~P(x).

Negation of Quantified Statements

~(∀x P(x)) ≡ ∃x ~P(x) ~(∃x P(x)) ≡ ∀x ~P(x)

When to use. Form the logical opposite of statements with quantifiers.

Trap. Do not forget to change the quantifier.

Rules of Inference

Modus Ponens: (p → q) ∧ p ⇒ q Modus Tollens: (p → q) ∧ ~q ⇒ ~p

When to use. Deduce conclusions from given premises.

Trap. Modus Ponens requires p to be true, not just p→q.

Necessary and Sufficient Conditions

p is sufficient for q: p ⇒ q p is necessary for q: q ⇒ p

When to use. Express relationships between conditions in statements.

Trap. Sufficient condition does not mean necessary.

Simplifying Compound Statements

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

When to use. Reduce complex logical expressions to simpler forms.

Trap. Do not confuse distributive laws with associative laws.

Read the full chapter: Mathematical Reasoning for JEE, with what JEE asks and the common traps.