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The five methods of proof

In essence there are five methods of proof:

  1. Direct: A string of implications A\implies B \implies \dots \implies Z, or some straightforward calculation.
  2. Induction: A(1) true and A(k)\implies A(k+1) true for all k means that A(n) is true for all n.
  3. Contradiction. Show a statement is true by assuming it is false and showing that this implies something that isn’t true.
  4. Contrapositive: To show A implies B we show B false implies A false.
  5. Cases: Divide the statement into separate cases. For example, x\le 0, x=0 and x\ge 0.

All proofs are constructed from these basic building blocks. In my first year course a key early aim is to make sure students know this. To achieve this I mark up some proofs during lectures noting the methods used (and where the assumptions are used, etc). They are then set the exercise of finding which methods of proof are used in a different example.

The exercise has a number of benefits. One, it tests the students on the methods. Two, it exposes the students to the structure of proofs. (An added benefit is that they are more likely to remember the proof when they know the structure.) Three, having explicitly seen the breaking down of proofs into pieces they are more likely to use that in the creation of their own proofs. (For the latter I have found that, generally, a prompt is needed, for example “how could we break it down?”)

As an example consider the following. To prove a statement of the form “P if and only if Q” students often try P\iff A\iff B\iff \dots \iff Q. That’s a great proof and it would be wonderful if it worked all the time but, unfortunately, generally it doesn’t and that causes many students to give up. However, if they have observed* repeatedly that P\iff Q can been divided into the cases P\implies Q and Q\implies P and those cases can be proved in different ways, e.g., one by direct method and the other by contradiction, then they are much more likely to use that method for attacking the problem. It’s always great when a student says “I can do the implication one way but not the other” — it shows that they’ve moved to a deeper level of understanding.

How do you show students the structure of proofs? Please comment below.

* By observed I mean they have seen it during something like the above exercise rather than merely told it is a possibility.

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