Abstract Algebra - Natural Numbers Proof

In summary, the conversation discusses the concept of sets of natural numbers that are closed under addition. The conversation mentions that odd numbers are not closed under addition, but sets of multiples are. The professor suggests picking numbers like 3 and 5 or 5 and 8 to expand the set and observe something not obvious. The conversation also mentions the Euclidean Algorithm and how it relates to the concept, as well as the regularity of the set. Lastly, the conversation concludes by stating that any natural number can be multiplied by an element in the set to get another element in the set, as long as n is large enough.
The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8, then expand the set.

For example 3,5 would be {3,5,6,8,9,10,11,12,13,14,15,...}

He said we should be able to observe over multiple sets something that is not completely obvious but I can't see anything. Possibly it has something to also do with the Euclidian Algorithm but I'm not so sure about that. Also, he said something about when the set starts to show regularity.

Any insight will help. Thanks!

I think you are on the right track. Note that you happened to choose two co-prime numbers. Do you see how at some point, you start to get all numbers (10, 11, 12, ...)?
Given 3 and 5, if you perform the Euclidean Algorithm, that basically gives you ... what?

Let n be any natural number

If a is in the set a*n must be
If a and b are in the set, gcd(a,b)*n is, if n is large enough.

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that deals with algebraic structures, such as groups, rings, and fields, and their properties. It is an abstract and general framework that allows mathematicians to study a wide range of mathematical systems.

2. What are natural numbers?

Natural numbers are the counting numbers, starting from 1 and increasing by 1 indefinitely. They are a subset of the whole numbers and include numbers such as 1, 2, 3, 4, etc.

3. What is a proof in abstract algebra?

A proof in abstract algebra is a logical argument that uses axioms, definitions, and previously proven theorems to show that a statement or proposition is true. In the context of natural numbers, a proof would involve using properties of natural numbers to demonstrate the truth of a given statement.

4. Why is it important to prove statements in abstract algebra?

Proving statements in abstract algebra is crucial because it allows us to establish the truth of mathematical statements and deepen our understanding of abstract mathematical structures. It also helps to build a foundation for further mathematical research and applications.

5. Can abstract algebra be applied to real-world problems?

Yes, abstract algebra has many practical applications in fields such as computer science, physics, and cryptography. For example, group theory is used in coding theory and cryptography, while ring theory is applied in coding theory and signal processing.

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