# Proof by Induction: Closure of Union = Union of Closures

• Bachelier
In summary, the conversation discusses the validity of using induction to prove that the closure of a union equals the union of the closures of the sets. The question of whether the proof is valid without being shown is raised, and the possibility of using induction for finite sets is mentioned. The conversation also delves into the concept of different styles of proof and the controversy around using induction for certain theorems. The conversation concludes with the sharing of an article on the Brouwer-Hilbert controversy.
Bachelier
Is it a solid proof to show that the closure of a union equals the union of the closures of the sets via induction?

Bachelier said:
Is it a solid proof to show that the closure of a union equals the union of the closures of the sets via induction?

How can we possibly know if your proof is valid if you don't show us your proof? Still, I'm not entirely sure why you would use induction here. Are you only considering the union of finitely many sets?

A proof is a proof no matter if you use induction or some direct method as long as the logic of each step is impeccable.

There are some math purists who disdain induction proofs for some theorems and prefer other styles of proof but that's beyond my math understanding to explain here:

http://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy

Number Nine said:
How can we possibly know if your proof is valid if you don't show us your proof? Still, I'm not entirely sure why you would use induction here. Are you only considering the union of finitely many sets?

Yes of course. I'm talking about finite sets. I found a different way to prove it though. But the proof by induction crossed my mind.

I should work it out some time and post it here. It seems feasible.

jedishrfu said:
A proof is a proof no matter if you use induction or some direct method as long as the logic of each step is impeccable.

There are some math purists who disdain induction proofs for some theorems and prefer other styles of proof but that's beyond my math understanding to explain here:

http://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy

great article. Thanks

## 1. What is "Proof by Induction: Closure of Union = Union of Closures"?

"Proof by Induction: Closure of Union = Union of Closures" is a mathematical concept that involves using the method of mathematical induction to prove that the closure of the union of two sets is equal to the union of their closures.

## 2. How does mathematical induction work?

Mathematical induction is a method of proving mathematical statements that involve natural numbers. It works by first proving that the statement is true for a base case (usually when n=1), and then assuming that the statement is true for some value of n and using that assumption to prove that the statement is also true for n+1. This process is repeated until the statement is proven to be true for all natural numbers.

## 3. Why is "Proof by Induction: Closure of Union = Union of Closures" important?

This concept is important because it is used to prove many mathematical statements involving sets. It also helps to understand the relationship between the closure of a union of sets and the union of their closures, which is a fundamental property in set theory.

## 4. Can you give an example of "Proof by Induction: Closure of Union = Union of Closures"?

One example is proving that the closure of the union of two closed sets is equal to the union of their closures. First, we prove that the statement is true for the base case of n=1 (i.e. when we have two closed sets A and B, the closure of their union is equal to the union of their closures). Then, we assume that the statement is true for some value of n and use that assumption to prove that it is also true for n+1 (i.e. when we have n+1 closed sets). By repeating this process, we can prove that the statement is true for all natural numbers, and thus, the closure of the union of any number of closed sets is equal to the union of their closures.

## 5. What are some applications of "Proof by Induction: Closure of Union = Union of Closures"?

This concept is used in many areas of mathematics, including abstract algebra, topology, and analysis. It is also important in computer science, as it is used to prove the correctness of algorithms and data structures.

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