Re: [math-fun] Theorem-dependance: possibly trivial question

There are a lot of theorem of the form The following are equivalent: Condition_1, ..., Condition_N. Occasionally the standard proof of such a theorem is of the form Condition_1 => Condition_2 => ... => Condition_N => Condition_1 . --Dan

On Mar 23, 2015, at 2:29 AM, Guy Haworth <g.haworth@reading.ac.uk> wrote:

Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then T1 --> T2.

Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting up loop T1 --> T2 --> T1) that could be explained to school children?

Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or cannot naturally, be shortened ? Schoolchildren-compatable preferred again !

Thanks - Guy