Simplify your problem by generalizing it

a repository of mathematical know-how

Simplify your problem by generalizing it

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]
One might think that generalizing a problem would make it harder to solve. However, the reverse is often true. The articles linked to from here explain why.

[GENERAL DISCUSSION]

[[Clarify your problem by making it more abstract]] [cut] Quick description || If you are asked to prove that a mathematical object $X$ has a property $P$, you will often obtain a clearer and easier problem if you identify certain properties $P_1,\dots,P_k$ of $X$ and then prove the statement "Every object that has properties $P_1,\dots,P_k$ has property $P$" instead of the original statement "$X$ has property $P$." See also [[Think axiomatically even about concrete objects]]. [/cut]

[[Strengthen your inductive hypothesis]] [cut] Quick description || Suppose that you are trying to prove by induction that a statement $P(n)$ is true for every $n$. If it is hard to deduce $P(n+1)$ from $P(n)$ then you may be able to deal with the difficulty by finding a different statement $Q(n)$ that implies $P(n)$, and deducing $Q(n+1)$ from $Q(n)$ instead (and, of course, proving $Q(1)$). [/cut]

[[To understand an object, consider treating it as one of a family of objects and analysing the family]] [cut] Quick description || An insight that has shaped much of modern mathematics is that it is often better to look at families of mathematical objects rather than at individual objects in isolation. If the family itself has a structure (usually geometric or algebraic) then it may be that this can be exploited: to prove a fact about one of the objects $X$ in the family it may be easier to prove a result about ''all'' the objects in the family and deduce from it what you want to know about $X$.[/cut]

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.

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.

Simplify your problem by generalizing it

Recent articles

Active forum topics

Recent comments