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Every proof strategy, and when to reach for it

Six ways to prove a claim -- read the claim's shape, then pick.

Read the claim before choosing a tool. A claim asserting something is impossible, or that nothing of a kind exists, almost always wants contradiction. A plain conditional wants direct proof if its hypothesis unfolds easily, or contrapositive if the negated conclusion unfolds more easily instead. A claim whose truth genuinely splits on a hidden condition (parity, sign, remainder) wants cases. An "exists" claim wants an existence proof -- constructive if a witness is easy to name, nonconstructive otherwise. And before committing to any of the above, a suspicious universal claim is always worth testing for a counterexample first.

Proving a conditional true

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Fix an arbitrary object satisfying the hypothesis, then chain justified steps to the conclusion.

Example

n even implies n squared even: let n=2k, then n squared = 2(2k squared).

Fix an arbitrary object satisfying the hypothesis, then chain justified steps to the conclusion.

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Prove the contrapositive instead, then invoke its guaranteed equivalence to the original.

Example

n squared even implies n even, proved by assuming n odd and showing n squared is odd.

Reach for this when the hypothesis is awkward to unfold but the negated conclusion is concrete.

Proving something is impossible, or false

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Assume the claim's negation, then derive an impossibility of the form R\wedge\neg R.

Example

No smallest positive rational exists -- assume one does, then r/2 is smaller.

Assume the negation, derive a statement of the form R and NOT R.

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A single object disproving a universal claim.

Example

n squared minus n plus 11 is prime fails at n=11.

Disproving a universal claim needs exactly one well-chosen witness that fails it.

When the truth splits on a hidden condition

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Split the domain into exhaustive cases, and prove the claim within each one.

Example

n squared plus n is always even -- one case for n even, one for n odd.

Split the domain into exhaustive cases (parity is the most common split), and prove each one.

Proving something exists (or exactly one exists)

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A proof establishing \exists x, P(x) -- constructively (naming a witness) or nonconstructively.

One verified witness is a complete proof -- no general argument needed.

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An existence proof that exhibits and verifies a specific witness.

Names and checks the witness directly.

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An existence proof that argues a witness must exist without ever naming one.

Argues a witness must exist without ever naming one -- pigeonhole (Chapter 10) is the classic example.

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A proof that any two objects satisfying a property must be equal.

A separate argument -- existence alone never confirms there is only one.

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