Chapter 15 — Appendix A1: Proofs in Mathematics

A mathematical statement is a meaningful sentence that is not an order, exclamation, or question, and is either always true or always false. Ambiguous sentences — those whose truth cannot be determined — are not accepted as mathematical statements. For example, 'The speed of light is approximately 3 × 10⁵ km/s' is always true, while 'Vehicles have four wheels' is ambiguous because vehicles can have 2, 3, 4, or more wheels.

Deductive reasoning is the main tool for constructing proofs. It involves drawing conclusions from given statements called premises or hypotheses. Each conclusion follows logically from the premises, from previously proved theorems, from axioms, or from definitions. Importantly, the correctness of the reasoning process does not depend on whether the hypotheses themselves are true — but an incorrect hypothesis may lead to a wrong conclusion.

A conjecture is an intelligent guess about a mathematical pattern or result. For example, from observing that joining n points on a circle with all possible lines creates 1, 2, 4, 8, and 16 regions for n = 1 to 5, one might conjecture that the number of regions is 2ⁿ⁻¹. However, for n = 6 there are only 31 regions (not 32), and for n = 7 there are 57 regions — so this conjecture is false. A conjecture requires a proof to be established as true.

It is enough to find a single counter-example — one specific case where the statement fails. For instance, the conjecture that n points on a circle create 2ⁿ⁻¹ regions is disproved by n = 6, which gives 31 regions instead of the predicted 32. In contrast, no number of verifications (for n = 1, 2, 3, 4, 5) can prove a statement true for all values of n.

The negation of a statement p is written ~p (read 'not p') and satisfies: ~p is false whenever p is true, and ~p is true whenever p is false. To negate, write the statement with 'not' — but be careful with statements involving 'All'. The negation of 'All teachers are female' is not 'All teachers are not female' (which could imply all are male), but rather 'There is at least one teacher who is not female.' The double negation ~(~p) always equals p.

A compound statement of the form 'if p, then q' (written p → q) has a converse 'if q, then p' (written q → p). p → q and q → p are converses of each other. A true statement's converse is not necessarily true. For example, 'If Ahmad is in Mumbai, then he is in India' is true, but its converse 'If Ahmad is in India, then he is in Mumbai' is false — he could be elsewhere in India.

To prove a statement p by contradiction: (1) Assume p is not true, i.e., assume ~p is true. (2) Using this assumption, carry out logical deductions. (3) If these deductions lead to a contradiction — a situation where some statement and its negation are both true — then the assumption ~p must be wrong. (4) Therefore, p must be true. This method was used in Chapter 1 to prove the irrationality of √2, and in this appendix to prove that the product of a non-zero rational number and an irrational number is irrational.

In a direct (deductive) proof, you start from the hypotheses and make a sequence of valid logical deductions until you reach the conclusion. In proof by contradiction, you assume the negation of what you want to prove, and show this assumption leads to a logical impossibility. Both methods are valid; as shown for Example 11 (every prime greater than 3 is of the form 6k + 1 or 6k + 5), different methods can prove the same result.

Theorem A1.1 (Converse of Pythagoras Theorem) states: if in a triangle the square of the length of one side equals the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. The proof is direct: given triangle ABC with AC² = AB² + BC², construct a right triangle ABD with BD = BC. Using the Pythagoras theorem, AD² = AB² + BD² = AB² + BC² = AC², so AC = AD. Then by SSS congruence, triangle ABC is congruent to triangle ABD, giving angle ABC = angle ABD = 90°.

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