# Set notation

1. May 18, 2009

### Bill Foster

I'm taking a class in abstract algebra this summer, so I thought I'd get ahead by reading the book before class starts.

This is from a book called "Abstract Algebra: A Geometric Approach", chapter 1:

Applying the Principle of Mathematical Induction with a slight modification.
If $$S' \subset \{n \in N:n\geq n_0\}$$ has these properties:
(1) $$n_0 \in S'$$
(2) If $$k \in S'$$ then $$k+1 \in S'$$
then $$S'=\{n \in N:n\geq n_0\}$$
If we define $$S=\{m \in N:m+(n_0-1) \in S'\}$$, we see that $$1 \in S$$ and $$k \in S$$, which leads to $$k+1 \in S$$ , and so $$S=N$$.
Thus, $$S'=\{n \in N: n=n_0+(m-1)$$ for some $$m \in N\}=\{n \in N:n \geq n_0\}$$

I'm not sure how to interpret all that. I know the sideways U means "subset", and the sideways U with a line means "is an element of". But does something like this $$\{n \in N:n\geq n_0\}$$ mean n is an element of N only when $$n \geq n_0$$?

What about this: $$S=\{m \in N:m+(n_0-1) \in S'\}$$?

How do you interpret that?

2. May 18, 2009

### tiny-tim

Hi Bill!

The : means "such that" …

so that means "S is the set of all elements m of N such that m + n0 - 1 is an element of S´"