1. Direct Proof & Counterexample

Video coming soon2 worked examples

Theory

A direct proof starts from the definitions and assumptions and deduces the conclusion through valid steps. The key is translating words into algebra: an even number is $2k$ , an odd number is $2k+1$ , for some integer $k$ .

To disprove a statement claimed to hold for all cases, a single counterexample is enough.

The Golden Rule: for a direct proof, turn the words into general algebra and manipulate to the required form. To disprove a “for all” claim, you need only one case that fails.

⚠️ Common Examiner Traps

Testing numbers instead of proving: checking a few cases does not prove a general statement — use a general $2k+1$ , etc.
Invalid counterexample: a counterexample must satisfy the hypothesis but break the conclusion.
Not reaching the required form: finish by showing the result is exactly what was asked (e.g. of the form $2 \times \text{integer}$ ).

Worked examples

Example 1

Prove that the sum of any two odd numbers is even.

Step 1: Let the two odd numbers be $2m+1$ and $2n+1$ , where $m, n$ are integers.

Step 2: Add them:

(2m+1) + (2n+1) = 2m + 2n + 2 = 2(m + n + 1)

Step 3: Since $m + n + 1$ is an integer, the sum is a multiple of $2$ , and is therefore even, as required.

Example 2

Disprove the statement: “ $n^2 - n + 41$ is prime for every positive integer $n$ .”

Step 1: Look for a value of $n$ that makes the expression factorise. Try $n = 41$ :

41^2 - 41 + 41 = 41^2 = 1681

Step 2: But $1681 = 41 \times 41$ , which is not prime. So $n = 41$ is a counterexample, and the statement is false.