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

Applying the Principle of Mathematical Induction with a slight modification.

If [tex]S' \subset \{n \in N:n\geq n_0\}[/tex] has these properties:

(1) [tex]n_0 \in S'[/tex]

(2) If [tex]k \in S'[/tex] then [tex]k+1 \in S'[/tex]

then [tex]S'=\{n \in N:n\geq n_0\}[/tex]

If we define [tex]S=\{m \in N:m+(n_0-1) \in S'\}[/tex], we see that [tex]1 \in S[/tex] and [tex]k \in S[/tex], which leads to [tex]k+1 \in S[/tex] , and so [tex]S=N[/tex].

Thus, [tex]S'=\{n \in N: n=n_0+(m-1)[/tex] for some [tex]m \in N\}=\{n \in N:n \geq n_0\}[/tex]

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 [tex]\{n \in N:n\geq n_0\}[/tex] mean

*n*is an element of

*N*only when [tex]n \geq n_0[/tex]?

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

How do you interpret that?