Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I found the following statement in Munkres' Topology:

Theorem 4.2 (Strong induction principle). Let $A$ be a set of positive integers. Suppose that for each positive integer $n$, the statement $S_n \subset A$ [here $S_n = \{1, 2, \dots, n\}$] implies the statement $n \in A$. Then $A = \mathbb{Z}_+$.

Now, I think I understand strong induction. But what I don't understand here is, doesn't $S_n \subset A$ always imply $n \in A$? It's pretty much by definition of a subset, that if $\{1,2,\dots,n\} \subset A$ then $n \in A$. Did the author mean "Suppose that for each positive integer $n$ the statement $S_n \subset A$ is true"?

share|improve this question
    
Unless, I'm missing something, I don't see how the statement is true unless $1 \in A$. –  user78680 May 20 '13 at 21:15
    
Since $S_1=\emptyset\subseteq A$, it follows that $1\in A$ by hypothesis. –  Cameron Buie May 20 '13 at 21:21
add comment

1 Answer

up vote 5 down vote accepted

If you check the previous page, you should see that $S_n$ is the set of positive integers that are less than $n$. $S_n$ here denotes the section of $\Bbb Z_+$ by $n$. See also the more general definition immediately before Lemma 10.2.

share|improve this answer
    
Oh, that makes a lot of sense. My bad. Thanks! –  Javier Badia May 20 '13 at 21:19
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.