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Learn/direct proofs/How Direct Proofs Work

Lesson subsection

How Direct Proofs Work

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

A direct proof is the most straightforward kind of proof for an implication:

If P is true, then Q is true.

The structure is:

  1. Assume P is true. (You literally say: "Let P be ...")
  2. Use definitions, known results, and algebra/logic to deduce new facts.
  3. Eventually reach Q as a conclusion.

The key mindset:

  • You are inside a world where P is true.
  • You are not arguing about whether P is true—you are using it as a starting point.
  • Every step must be justified by a definition, a known theorem, or a clear logical rule.

Example pattern:

Prove: If n is an even integer, then n² is even.

  • Assume n is even ⇒ n = 2k for some integer k.
  • Then n² = (2k)² = 4k² = 2(2k²), which is an even number.
  • Therefore, if n is even, n² is even.

This is a direct proof: we assumed the hypothesis and algebraically reached the conclusion.

TL;DR — key idea

A direct proof of "If P, then Q" starts by assuming P, logically deduces facts using definitions and known results, and ends by showing Q must follow.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

  • 1

    In your own words, describe the three main steps of a direct proof for a statement "If P, then Q". Then choose a simple example (e.g., "If n is divisible by 4, then n is even") and outline the structure of its direct proof without filling in all the algebra.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.