Understanding Abstract Algebra: A Geometric Approach

[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?

 

[tiny-tim]

… 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?

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´"