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

a repository of mathematical know-how

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

Title: *

Area of mathematics: *

A comma-separated list of areas of mathematics to which this article applies. Use ">" to tag in a subcategory. Example: Analysis > Harmonic analysis, Combinatorics

Keywords:

A comma-separated list of keywords associated with this article. Example: free group

Used in:

A comma-separated list of examples of where this technique is used. Example: Cauchy-Schwarz inequality

Parent articles:		Order

Body:

[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| Isosceles Triangles]
[image Triangle_1.gif|Isosceles triangle]
[theorem isosc]If a triangle has two equal angles, then it has two equal sides.[/theorem]
[proof isosc]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.[/proof]

[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 [[w: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|Taking the inverse is an involution]

[definition invdef] In a set with an associative binary operation and an identity element $e$, an element $y$ is the '''inverse''' of an element $x$ if 
[maths invdef]xy = e \text{ and } yx = e[/maths][/definition]

In this situation, it is easy to see that $x$ has only one inverse. 
 
[theorem invinv] $(x^{-1})^{-1} = x$.[/theorem]

[proof invinv]We are given that $x^{-1}$ is the inverse of $x$.  By [ref definition #invdef], this means that 
[maths invfact]xx^{-1} = e \text{ and } x^{-1}x = e[/maths]           
To prove [ref theorem #invinv], we must show that $x$ is the inverse of $x^{-1}$, which requires that  
[maths invnec]x^{-1}x = e \text{ and } xx^{-1} = e[/maths]
But [eqref invfact] and [eqref invnec] are equivalent!
[/proof]

[GENERAL DISCUSSION]

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

[note article incomplete] This article needs some more sophisticated examples [/note]

This is a stub

A stub is an article that is not sufficiently complete to be interesting.

Notifications

File attachments

Changes made to the attachments are not permanent until you save this post. The first "listed" file will be included in RSS feeds.

	Delete	List	Description	Weight	Size
			/images/Triangle_1.gif		883 bytes

Attach new file:

Images are larger than 640x480 will be resized. The maximum upload size is 1 MB. Only files with the following extensions may be uploaded: jpg jpeg gif png svg.

Revision information

Log message:

An explanation of the additions or updates being made to help other authors understand your motivations.

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

Recent articles

Active forum topics

Recent comments