## October 27, 2004

### An Exercise for Learning Proofs

The main use of learning Euclidean geometry is to learn the idea of how to prove things-- how to go logically from assumptions to a proposition. That is especially useful because the assumptions are laid out as axioms, but doing any kind of proof can be a useful exercise. Below is an exercise that I think I heard about from Professor Aliprantis some years back:...

... Ask students to prove that adding some numbers together to equal 5, at least one number must be greater than 2.

Part of the task is to clarify the proposition. It takes some thought to come up with the following statement of the problem:

PROPOSITION 1: If x+y =5, where x and y are positive integers, either x or y is greater than 2.

Having proved that, have the students prove Proposition 2.

PROPOSITION 2: If x+y = 101, where x and y are positive integers, either x or y is greater than 50.

This is a good second step because Proposition 1 can be proved by exhaustion.

Posted by erasmuse at October 27, 2004 09:40 AM