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.

 

[CellCoree]

Mathematical induction is a powerful mathematical tool used to prove statements about a set of numbers or objects. It is particularly useful in the field of set theory and relations, where it allows us to prove properties of infinite sets by using a finite number of steps.

The main idea behind mathematical induction is that if we can show that a statement holds for a specific number or object, and then prove that if the statement holds for that number or object, then it also holds for the next number or object, then we can conclude that the statement holds for all numbers or objects in that set.

In the context of set theory and relations, mathematical induction allows us to prove properties of sets that have an infinite number of elements. For example, we can use mathematical induction to prove that all positive integers are divisible by 3, or that every set has a subset of smaller cardinality.

There is a deep relationship between mathematical induction and other mathematical concepts such as sets, relations, and functions. In fact, mathematical induction can be seen as a special case of the more general concept of proof by induction, which is used in many branches of mathematics. Mathematical induction is also closely related to the concept of recursion, where a function is defined in terms of itself.

In conclusion, mathematical induction is a fundamental concept in mathematics, particularly in the fields of set theory and relations. It allows us to prove properties of infinite sets using a finite number of steps, and it is closely related to other important mathematical concepts.