0

Define $S=\left\{ an+b:n\in \mathbb{N}\right\}$, where a and b are integers and $a\neq0$.

Write down the statement of ordinary induction on S.

What does it mean by "statement of ordinary induction on S" Isn't induction a proving method? What do they mean by statement?

Moko19
  • 2,625
diamondapple123
  • 87
  • I'd guess they meant something like "show that $S$ has a least element and that there is a natural notion of successor in $S$". – lulu Sep 08 '22 at 15:27
  • 2
    They mean that you should formulate the induction method allowing to prove that some statement is true for every element of the set $S$. – Peter Sep 08 '22 at 15:28
  • Depending whether $0\in \mathbb N$ or not, the start is $b$ or $a+b$. Given that the statement is true for some natural number $n$ , you have to show that it is also true for $n+a$ which finishes the proof. – Peter Sep 08 '22 at 15:30
  • 1
    Maybe they mean: For any statement $P$ if $P(b)$, and if [$P(an + b) \implies P(a(n+1)+b)$], then $P(x)$ for all $x \in S$. – fleablood Sep 08 '22 at 15:31
  • @fleablood A good point. I just noticed that my version might fail, for example if the statement holds ONLY for numbers of the form $an+b$. – Peter Sep 08 '22 at 15:34
  • 3
    By the way, OP, it's worth mentioning that I too find this instruction to be unclear...! (I can guess what it means because of long experience, but the task should be worded more clearly if students are meant to understand it.) – Greg Martin Sep 08 '22 at 15:44
  • Where is this problem taken from? – Moko19 Sep 08 '22 at 16:50

1 Answers1

0

I think they probably mean the following although I've never seen something stated like this;

$$an+b=m\implies a(n+1)+b=an+b+a=m+a$$

$$\therefore S(n)=m \implies S(n+1)=m+a$$

Volk
  • 1,805