Direct proof

Objective. Construct direct proofs from definitions and hypotheses.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Direct proof establishes a statement P⇒Q by assuming P is true and using definitions, algebra and known results to deduce Q logically. Each line must follow from the previous one or from an accepted fact, with no gaps. A classic pattern: to prove "if n is odd then n² is odd", write n=2k+1 for some integer k, then compute n²=4k²+4k+1=2(2k²+2k)+1, which has the form 2m+1, hence is odd. The key discipline is unpacking definitions (odd means n=2k+1) rather than relying on intuition. Direct proofs are preferred whenever the forward chain of implications is clear and manageable.

Worked example

Give a direct proof that if n is an even integer, then n² is even.

1 $Assume n is even, so by definition n=2k for some integer k.$
2 $Compute n²=(2k)²=4k².$
3 $Rewrite 4k²=2(2k²), which has the form 2m with m=2k² an integer.$
4 $Since n²=2m, n² is even, completing the direct proof.$

Answer: n²=2(2k²), which is even

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common error is proving a specific numerical example (like checking n=3) and treating it as a general proof; one example never establishes a universal claim. A direct proof must use an arbitrary representative, such as n=2k+1 for unspecified integer k.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard