Practice question · Put in order
Order the steps of a direct proof that the sum of two even integers is even.
- Write m = 2a and n = 2b for some integers a and b
- Assume m and n are even integers
- Conclude m + n is even
- Note that a + b is an integer
- Add them: m + n = 2a + 2b = 2(a + b)
Hints
- A direct proof starts by assuming the hypothesis and ends at the conclusion.
- The definition of 'even' must be unpacked before you can compute, and re-applied at the end.
Show the answer
- Assume m and n are even integers
- Write m = 2a and n = 2b for some integers a and b
- Add them: m + n = 2a + 2b = 2(a + b)
- Note that a + b is an integer
- Conclude m + n is even
Why
A direct proof runs hypothesis to conclusion: assume, unpack the definition into algebra, compute, check the result fits the definition again, conclude. Step 4 matters, 2(a+b) only proves evenness because a+b is itself an integer.
Practise Direct Proof and Counterexample
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