Mathematical Proof Examples – Exam-Style Questions
The Proof Examples collection is a favourite in StudyWell’s collection of downloadable resources (see more downloadable resources). There are 16 exam-style examples in the Mathematical Proof Collection (16 statements to prove or disprove in total) covering proof by deduction, proof by exhaustion and disproof by counterexample.
The solutions document shows the statements and their respective proofs or disproofs. The explanations are detailed and clear. You can read this PDF on-the-go from your smartphone or download it to an iPad or desktop.
Hints for tackling the Mathematical Proof Examples
For each statement in the Proof Examples collection you must first decide whether the statement is true or not. If you believe the statement to be untrue then find a single example when the statement fails (Disproof by Counterexample). Alternatively, if you think the statement is true then decide whether to prove it by Proof by Deduction or Proof by Exhaustion. If you are unsure, then you might like to try testing the statement with various examples first. Be sure to try the statement with negative numbers as well as positive ones. However, note that showing that the statement is true for a few examples does not prove it – see common mistakes below.
When attempting to complete the Mathematical Proof Examples, there are a number of common mistakes that students tend to make. Here are the top three:
- For the Mathematical Proof Examples that test students’ ability to perform Proof by Deduction, students often give answers that demonstrate particular examples. We do not show that something is always true by showing that it is true for a few example numbers – we must show that it is always true using algebra. See Proof by Deduction.
- Students often make the mistake of assuming something is true to show that it is true. This usually results in circular arguments that make no sense.
- Another common mistake made when answering Mathematical Proof Examples is the misuse of algebra. For example, $(n+1)^2$ is often mistakingly expanded as $n^2+1$. Of course, it should be $n^2+2n+1$ which we obtain by expanding the double brackets $(n+1)(n+1)$.