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Recognize that the same object fits a pattern in two different ways

Quick description

Substituting the same mathematical object in two different ways into the same pattern can give a quick proof of a fact.

Prerequisites

Elementary.

Example 1: Isosceles Triangles

Isosceles triangle

Theorem 1 If a triangle has two equal angles, then it has two equal sides.

Proof of Theorem 1. In the figure, assume \angle ABC = \angle ACB. Then triangle ABC is congruent to triangle ACB since the sides BC and CB are equal (they are the same line segment!) and the adjoining angles are equal by hypothesis.

General discussion

The point is that although triangles ABC and ACB are the same triangle and sides BC and CB are the same line segment, the proof involves recognizing them as geometric figures in two different ways.

This proof is over two millenia old and is called the Pons_asinorum(bridge of donkeys). It became famous as the first theorem in Euclid's books that many students could not understand. I conjecture that the name comes from the fact that the triangle as drawn here resembles an ancient arched bridge. Usually isosceles triangles are drawn taller than they are wide.

Example 2: Taking the inverse is an involution

Definition 1 In a set with an associative binary operation and an identity element e, an element y is the inverse of an element x if
xy = e \text{ and } yx = e(1)

In this situation, it is easy to see that x has only one inverse.

Proof of Theorem 2. We are given that x^{-1} is the inverse of x. By definition 1, this means that

xx^{-1} = e \text{ and } x^{-1}x = e(2)

To prove theorem 2, we must show that x is the inverse of x^{-1}, which requires that

x^{-1}x = e \text{ and } xx^{-1} = e(3)

But (2) and (3) are equivalent!

General discussion

In this example, we have substituted the variables x and x^{-1} into the same equation in two different ways.

Note iconIncomplete This article is incomplete. This article needs some more sophisticated examples

Comments

Courtesy plural

Instead of saying "I am given..." and "I must show...", I would use the "courtesy plural" and say "We are given..." and "We must show..." as this is the standard rule for scientific writing, If I'm not mistaken ;-)

Courtesy plural

I will change this, but as I understand the way a wiki works, you could have changed it yourself. The author doesn't own the article.

Editing vs good writing

Yes, you are right; I could have done the changes myself and I usually do this kind of work in Wikis (format/grammar editing). But here we have the luck of being a small site (yet) and I had the opportunity to expose my opinion directly to you and, in case you got convinced of the correctness of the change, I would have saved myself from the need of editing all your articles anytime you didn't use the courtesy plural ;-)

It's just the old topic of teaching how to grow wheat instead of just giving bread, if you allow me the comparison.