What is the use of mathematical induction

[pairofstrings]

In the context of Set theory and Relations why do we use mathematical induction. Is there any deep relation between all these concepts or mathematical induction is only a separate concepts introduced in the textbooks after Sets and Relation ; Functions ; and then Mathematical induction.

 

[micromass]

It would be nice of you if you could expand your question a bit, because I'm not sure what you're getting at here. How did you use induction in sets and relations?

Anyways, induction is a really useful tool for proving things for natural numbers (or more generally: for well-ordered sets). In fact, the tool of induction is so important that it characterized the natural numbers in some way. That is, if we didn't have induction available, then the natural numbers wouldn't be what we expect them to be. This is reflected in the Peano axioms, where induction is taken to be one of the crucial axioms of Peano arithmetic.

So induction is not only useful, it is necessary if you want to prove anything important for natural numbers.

Of course, induction for natural numbers can be extended to transfinite induction which works over well-ordered sets. In the context of set theory, this is of extreme importance. It allows you to prove results like Zorn's lemma, who's use is well-documented...

 

[zketrouble]

Could you give a little more detail about your question? Its a bit too vague for me to understand exactly what you're asking.

 

[Landau]

A low-level answer: it allows us to do recursion. E.g. define the factorial as

0!=1
n!=(n-1)! for all n>0.