Once we can read and combine statements, the central question of the topic is: how do we know a statement is true? Validating a statement means proving it, and the form of the statement decides the method. There are four standard techniques in the syllabus.
1. The direct method. To prove "if $p$ then $q$", assume $p$ is true and, through valid steps, deduce that $q$ is true. This is the most common method and matches the natural flow of an argument. To prove a "for all" statement you argue with an arbitrary (general) element so the conclusion holds for every case, not just an example.
2. The method of contrapositive. Because $p \Rightarrow q$ is logically equivalent to $\sim q \Rightarrow \sim p$, you may instead assume $\sim q$ and deduce $\sim p$. This is invaluable when the negation $\sim q$ is easier to work with than $q$ itself — for example, statements about "odd/even" or "irrational" are often cleaner from the contrapositive end.
3. The method of contradiction (reductio ad absurdum). To prove a statement $p$, assume the opposite — that $\sim p$ is true — and show this assumption forces a logical contradiction (something known to be false, such as $a = b$ and $a \ne b$ together). Since the assumption leads to an absurdity, it must be wrong, so $p$ is true. The classic example is the proof that $\sqrt{2}$ is irrational: assuming it is rational ($\sqrt{2} = p/q$ in lowest terms) leads to $p$ and $q$ both being even, contradicting "lowest terms".
4. Disproof by counter-example. The previous three methods establish truth. To disprove a "for all" statement, you do not need a general argument — a single counter-example is enough, because one exception destroys a universal claim. Note the asymmetry: a counter-example can disprove but can never prove a "for all" statement, since checking finitely many cases never covers all of them.
| Goal | Method | You assume… |
|---|---|---|
| Prove $p \Rightarrow q$ | Direct | $p$, then deduce $q$ |
| Prove $p \Rightarrow q$ | Contrapositive | $\sim q$, then deduce $\sim p$ |
| Prove $p$ | Contradiction | $\sim p$, reach absurdity |
| Disprove $\forall$ claim | Counter-example | nothing — exhibit one exception |
For a biconditional "$p$ if and only if $q$", validity requires proving both directions separately: $p \Rightarrow q$ and $q \Rightarrow p$. Proving only one direction is a common and serious gap.
Deeper Insight — match the method to the shape of the claim: The skill being trained here is not memorising four recipes but diagnosing which one fits. Ask what kind of statement you face. A conditional invites the direct method first; if the conclusion is awkward but its negation is clean, switch to the contrapositive — they are the same proof seen from opposite ends. An existence or "no such object" claim, or anything where assuming the opposite gives you something concrete to manipulate, suits contradiction. And the moment you suspect a universal claim is false, stop trying to prove it and hunt for one counter-example. Two warnings carry most of the marks: a counter-example never proves a universal statement, and an "if and only if" demands both directions. Internalise the pairing between claim-shape and method, and validation stops being guesswork.