Common Structures of Arguments

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

Most proofs are not random collections of sentences; they follow common patterns.

Some core structures you will see often:

Direct proof: Assume the hypothesis, then logically derive the conclusion.
Contrapositive: Prove 'if not Q then not P' instead of 'if P then Q'.
Proof by contradiction: Assume the statement is false and derive an impossibility.
Case analysis: Split into cases and prove the statement in each case.

Recognizing these structures turns proofs from mysterious paragraphs into familiar templates you can apply and adapt.

TL;DR — key idea

Proofs follow standard patterns like direct proof, contrapositive, contradiction, and casework. Learning these templates makes proofs far less intimidating.

Try these in your notebook

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

1
Write a one-sentence description of direct proof, contrapositive, and contradiction in your own words.
2
Pick any simple 'if P then Q' statement (for example, 'if n is even then n^2 is even') and identify which proof structures could be used.

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.