I'm a beginner. According to my professor, I must write out every step as clearly as possible.
Question: Prove the set of positive even integers is well-ordered using the well-ordering principle.
Solution (According to Bartelby.com): By definition, the set of positive even integers is the subset of positive integers. Moreover, since the even integers are well-ordered, (by the well-ordering principle), the positive even integers are well-ordered.
However, I'm unable to understand the proof. How do we know that if a set has the property then its subset has the property? (My Professor demands I rely on proofs rather than intuition).
Below is my attempt at the problem:
My attempt: By definition, the set of positive even integers is the subset of positive even integers. Hence if the set of positive even integers are well-ordered, then; the set of postive integers are well-ordered. Despite this, we must prove if positive integers are well-ordered then positive even integers are well-ordered. Therefore, we take the contrapositive.
If the positive integers are not well-ordered then the positive even integers are not well-ordered; however, the positive integers are well-ordered (by the well-ordering principle). This is a contradiction to our hypothesis where the positive integers are not well-ordered. Hence if the positive integers are well-ordered then the positive even integers are well-ordered. Therefore, the positive even integers are well-ordered.
Question: Am I correct or have I made a mistake in my steps? Try and explain step by step?
Best Answer
You can refer to the definition of a well ordered set to prove that any subset of a well ordered set is well ordered. A well ordered set is one that any subset has a smallest element. If you take a subset of a well ordered set, all of its subsets are subsets of the larger set, so they have a smallest element. This shows the subset is well ordered.