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Q:

S is a subset of the natural numbers (the counting numbers) with the following property

n [itex]\in[/itex] S [itex]\Rightarrow[/itex] {1,...,n-1} [itex]\subset[/itex] S.

Prove, by Reductio Ad Absurdum, that S is equal to the set of all natural numbers.

First of all, I want to make sure I got all the meanings of the symbols right, so are the following correct?

- n [itex]\in[/itex] S means n is an element of the set S.

- {1,...,n-1} [itex]\subset[/itex] S means every element of {1,...,n-1} is also an element of S, but {1,...,1-n} ≠ S

So that means S must contain more elements than {1,...,n-1} does, right? (i.e. S can be {1,...,n-1,n} ??)

If the above statements are correct and I'm performing a proof by contradiction, how should I start the first sentence?

I know that if I want to prove "1/3 is rational", I'll start by assuming the statement is false, i.e. I'll assume that 1/3 is irrational.

But this question is too complicated, if I am to assume that "n [itex]\in[/itex] S [itex]\Rightarrow[/itex] {1,...,n-1} [itex]\subset[/itex] S" is false, how should I write it in words?

S is a subset of the natural numbers (the counting numbers) with the following property

n [itex]\in[/itex] S [itex]\Rightarrow[/itex] {1,...,n-1} [itex]\subset[/itex] S.

Prove, by Reductio Ad Absurdum, that S is equal to the set of all natural numbers.

First of all, I want to make sure I got all the meanings of the symbols right, so are the following correct?

- n [itex]\in[/itex] S means n is an element of the set S.

- {1,...,n-1} [itex]\subset[/itex] S means every element of {1,...,n-1} is also an element of S, but {1,...,1-n} ≠ S

So that means S must contain more elements than {1,...,n-1} does, right? (i.e. S can be {1,...,n-1,n} ??)

If the above statements are correct and I'm performing a proof by contradiction, how should I start the first sentence?

I know that if I want to prove "1/3 is rational", I'll start by assuming the statement is false, i.e. I'll assume that 1/3 is irrational.

But this question is too complicated, if I am to assume that "n [itex]\in[/itex] S [itex]\Rightarrow[/itex] {1,...,n-1} [itex]\subset[/itex] S" is false, how should I write it in words?

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