Mathematical Reasoning Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 6 subtopics

Negation of a Statement

\sim p \text{ is true when } p \text{ is false, and vice versa}

💡 Negation flips the truth value. If p is 'x > 5', then ~p is 'x is not greater than 5' (i.e., x <= 5).

Conjunction (AND)

p \wedge q \text{ is true only when both } p \text{ and } q \text{ are true}

💡 AND requires both parts to be true. Even one false component makes the entire conjunction false.

Disjunction (OR)

p \vee q \text{ is false only when both } p \text{ and } q \text{ are false}

💡 OR is inclusive in logic. It is true when at least one component is true.

Conditional (If-Then)

p \to q \equiv \sim p \vee q

💡 A conditional is false ONLY when p is true and q is false. 'If it rains, the ground is wet' is false only when it rains but the ground is dry.

⚠ Students think p -> q is false when p is false. A conditional with a false hypothesis is ALWAYS true (vacuously true).

Contrapositive

p \to q \equiv \sim q \to \sim p

💡 The contrapositive always has the same truth value as the original conditional. Converse (q -> p) and inverse (~p -> ~q) do NOT.

⚠ Confusing contrapositive with converse. Contrapositive: negate and swap. Converse: just swap (not equivalent).

Biconditional (If and Only If)

p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)

💡 Biconditional is true when both p and q have the same truth value (both true or both false).

De Morgan's Laws for Logic

\sim(p \wedge q) \equiv \sim p \vee \sim q \quad | \quad \sim(p \vee q) \equiv \sim p \wedge \sim q

💡 Negation of AND becomes OR (with negated parts). Negation of OR becomes AND (with negated parts). Swap the connective and negate each component.

Negation of Quantifiers

\sim(\forall x, P(x)) \equiv \exists x, \sim P(x) \quad | \quad \sim(\exists x, P(x)) \equiv \forall x, \sim P(x)

💡 Negation of 'for all' becomes 'there exists ... not'. Negation of 'there exists' becomes 'for all ... not'. Swap the quantifier and negate the predicate.

Principle of Mathematical Induction

P(1) \text{ is true} \wedge [P(k) \text{ true} \Rightarrow P(k+1) \text{ true}] \implies P(n) \text{ true } \forall \, n \in \mathbb{N}

💡 Two steps: (1) Base case: verify P(1). (2) Inductive step: assume P(k) and prove P(k+1). Both steps are mandatory.

⚠ Forgetting to verify the base case. Without it, the induction hypothesis has no foundation.

Strong Induction

#10

[P(1) \wedge P(2) \wedge \ldots \wedge P(k) \Rightarrow P(k+1)] \implies P(n) \text{ true } \forall \, n

💡 In strong induction, assume P(m) is true for ALL m from 1 to k, then prove P(k+1). Useful when P(k+1) depends on multiple previous cases, not just P(k).