Practice question · Put in order
Order the steps of a direct proof that if n is even then n squared is even.
- Square both sides to get n squared = 4k squared
- Rewrite 4k squared as 2 times (2k squared), an integer doubled
- Conclude that n squared is even, as required
- Assume the hypothesis: n is even
- Write n = 2k for some integer k, which is what being even means
Hints
- A direct proof runs from the hypothesis to the conclusion, so it starts by assuming what it is given.
- The definition of even has to be unpacked before it can be used, and repacked before the conclusion can be stated.
Show the answer
- Assume the hypothesis: n is even
- Write n = 2k for some integer k, which is what being even means
- Square both sides to get n squared = 4k squared
- Rewrite 4k squared as 2 times (2k squared), an integer doubled
- Conclude that n squared is even, as required
Why
A direct proof assumes the hypothesis, converts definitions into usable algebra, manipulates, and converts back. The final rewriting step matters: 4k squared is not obviously even until it is displayed as 2 times an integer, which is exactly the definition being claimed.
Practise Proof Techniques
The app has 6 more questions on this lesson, and keeps your place in the course. Computer Science I is free to start.
More questions on Proof Techniques
- A proof by contradiction and a proof by contrapositive both begin by assuming something false. Which is…
- A student proves "if n² is even then n is even" by assuming n is odd, writing n = 2k+1, and deriving that n²…
- The claim is: for every integer n, if n squared is odd then n is odd. Select every approach that would…