1st PUC Mathematics · Chapter 14

Mathematical Reasoning — The Logic Behind Every Proof You’ll Ever Write

Mathematical Reasoning is a short chapter with an outsized reputation for confusion — mostly because students memorise the terminology (conjunction, disjunction, negation) without connecting it to how they already reason in everyday language. Every rule here matches ordinary logic; the chapter just gives it precise names and symbols.

What Counts as a Statement

A mathematical statement is a sentence that is either definitely true or definitely false — not both, and not ambiguous. “Delhi is the capital of India” is a statement (true). “Please close the door” is not a statement — it’s a command, with no true/false value. “x + 2 = 5” is not a statement on its own either, since its truth depends on the unknown value of x — it only becomes a statement once x is specified.

The Connectives

Conjunction (AND, ∧): “p ∧ q” is true only when both p and q are true. “It is raining and it is cold” is true only if both conditions hold.

Disjunction (OR, ∨): “p ∨ q” is true when at least one of p, q is true — and true when both are true too. This is the “inclusive or” used throughout mathematics, different from the everyday “either/or” that sometimes implies exclusivity.

Negation (NOT, ~): “~p” flips the truth value — if p is true, ~p is false, and vice versa. Negating compound statements follows specific rules: ~(p ∧ q) = ~p ∨ ~q, and ~(p ∨ q) = ~p ∧ ~q — known as De Morgan’s Laws, and one of the most directly testable results in this chapter.

Implications and Their Variants

“If p then q” (p ⟹ q) is false only in one specific case: when p is true but q is false. In every other combination of truth values, the implication itself is true — including, counterintuitively, when p is false regardless of q. This is the single most misunderstood rule in the chapter, and worth sitting with rather than just memorising.

Three related statements follow from any implication p ⟹ q: the converse (q ⟹ p), the inverse (~p ⟹ ~q), and the contrapositive (~q ⟹ ~p). Only the contrapositive is guaranteed to have the same truth value as the original implication — the converse and inverse can independently be true or false.

Worked Example — Negating a Compound Statement

Negate: “The number is even and divisible by 3.” Using De Morgan’s Law on the conjunction: the negation is “The number is not even OR it is not divisible by 3” — note the AND becomes OR, a detail students frequently miss when negating by intuition instead of applying the rule.

Where Students Actually Lose Marks

The most common error is forgetting that AND flips to OR (and vice versa) when negating a compound statement via De Morgan’s Law. The second is confusing the converse with the contrapositive — only the contrapositive is logically equivalent to the original statement; the converse is a genuinely different claim that needs its own proof.

Frequently Asked Questions

Why is “if p then q” true when p is false?

Because the implication only makes a promise about what happens if p is true — when p never happens, the promise is never broken, so the implication is vacuously true. This matches how the rule is defined in formal logic across all of mathematics.

Is the contrapositive really always equivalent to the original statement?

Yes — this is a fixed logical fact, not something that depends on the specific statement, which is why proof by contrapositive is a valid proof technique used throughout later mathematics.

Does this chapter connect to proof-writing in later chapters?

Yes — the logic here directly underlies how proofs in Mathematical Induction and later 2nd PUC chapters are structured and validated.

Logic You Already Use, Made Precise

This chapter is shorter than most, but the reasoning skills it builds show up everywhere afterward. Daniel Sir connects it to everyday logic first, then to the formal rules. See the PUC Mathematics Programme →