Contradiction and rational numbers

I have a set that is closed under addition and multiplication. Additionally, for any , one of the three is true

I have to prove that all positive integers are in .

So I assume there exists a positive integer , hence .

Not sure which direction to go in to get the contradiction.

Help please?

Re: Contradiction and rational numbers

Re: Contradiction and rational numbers

Quote:

Originally Posted by

**I-Think** I have a set

that is closed under addition and multiplication. Additionally, for any

, one of the three is true

I have to prove that all positive integers are in

.

So I assume there exists a positive integer

, hence

.

Not sure which direction to go in to get the contradiction.

Help please?

You **can't** prove it. It is not true.

For example S= {0} satisfies all of the conditions but does not include all of the integers.

You need some additional condition, perhaps the "1 in S" that Plato suggested.

Re: Contradiction and rational numbers

Quote:

Originally Posted by

**I-Think** Additionally, for any

, one of the three is true

Quote:

Originally Posted by

**HallsofIvy** For example S= {0} satisfies all of the conditions but does not include all of the integers.

I don't think S = {0} satisfies .

Re: Contradiction and rational numbers

Quote:

Originally Posted by

**emakarov** I don't think S = {0} satisfies

.

You are right. I was thinking the condition was " ".