Prove a consequence first!

Quick description

If one wants to show “If A, then B”, first find an interesting consequence C of B that looks easier to prove than B itself, but not so simple that it can be immediately deduced from A. Then try to prove “If A, then C”. Finally, show “If A and C, then B”. The point is that this factors the original problem into two simpler ones. If one believes that the original implication is true, then the two sub-implications must be true also, so one is not “losing” anything by trying this method.

A variant approach, once one has successfully obtained “If A, then C”, is to deconstruct that proof in order to find out what made it work; this can then give valuable clues as to how to then prove the stronger statement “If A, then B”. This tends to work well when C acts as a “simplified toy model” of B.

This post is based on this Google Buzz article by Terence Tao

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s