Set Notation Question: Converting to Set Builder Notation | Homework Help

[Rijad Hadzic]

Homework Statement

So I have the set

{...(1/8),(1/4),(1/2),(1),(2),(4),(8)...}

I am suppose to put it in set builder notation..

Homework Equations

The Attempt at a Solution

my answer was {x = {\frac {1}{2^n} : n \in ℤ }}

but my books was

{x = {{2^n} : n \in ℤ }}

I understand both answers to be true. But would my answer be valid say, on a test or something. Is one of these preferred over the other?

 

[Mark44]

Homework Statement

So I have the set

{...(1/8),(1/4),(1/2),(1),(2),(4),(8)...}

I am suppose to put it in set builder notation..

Homework Equations

The Attempt at a Solution

my answer was {x = {\frac {1}{2^n} : n \in ℤ }}

but my books was

{x = {{2^n} : n \in ℤ }}

I understand both answers to be true. But would my answer be valid say, on a test or something. Is one of these preferred over the other?

In any but a brain-dead computerized quiz, both should be recognized as correct. IMO, neither one would be preferred over the other.

 

[Rijad Hadzic]

In any but a brain-dead computerized quiz, both should be recognized as correct. IMO, neither one would be preferred over the other.

Okay thank you. I just wanted to make sure..

 

[Ray Vickson]

Okay thank you. I just wanted to make sure..

The reason they are equivalent is that
$$ 2^{-n} = \frac{1}{2^n}, \: \text{and} \; 2^n = \frac{1}{2^{-n}}, $$
so that when $n$ runs through all positive and negative integers, for every $n \in \mathbb{Z}$ value of $2^n$ is matched exactly by $1/2^m$, where $m \in \mathbb{Z}$.