Infinite Subsets: Length & Terms Explained

[jason177]

Alright this is a pretty simple question. So if A is a infinite subset of the reals, does that mean that its length is infinite (eg. the set of all numbers from 0 to infinite) or could it also be just a set with an infinite number of terms (eg. the set of all real numbers between 1 and 2)?

 

[SammyS]

Alright this is a pretty simple question. So if A is a infinite subset of the reals, does that mean that its length is infinite (eg. the set of all numbers from 0 to infinite)

No.

or could it also be just a set with an infinite number of terms (eg. the set of all real numbers between 1 and 2)?

Yes.In fact its length could be zero.

 

[Robert1986]

Alright this is a pretty simple question. So if A is a infinite subset of the reals, does that mean that its length is infinite (eg. the set of all numbers from 0 to infinite) or could it also be just a set with an infinite number of terms (eg. the set of all real numbers between 1 and 2)?

Everyone's favorite counter example to everything: The Cantor Set.

Let C be the Cantor set. C is a subset of the unit interval. After you remove C from the unit interval, the length of what is left in the unit interval is 1, so the length of C is 0. At the same time, there are the same number of points in C as there are left in the unit interval, so C is infinite. C is an odd little set, you should check it out.

 

[jason177]

alright, thank you both for your replies