Understanding Zero Sets: Real Analysis Examples

[Demon117]

What is the definition of a zero set and what exactly does it mean?

I have come across different responses on the internet, but none of them explain really what it means or give good examples, I am having a rough time with this concept in real analysis.

For example, how would I determine if {(x,f(x)) : x in R} (f maps R to R is continuous) is a zero set? What would I be looking at to help determine that?

 

[gb7nash]

What is the definition of a zero set and what exactly does it mean?

I have come across different responses on the internet, but none of them explain really what it means or give good examples, I am having a rough time with this concept in real analysis.

For example, how would I determine if {(x,f(x)) : x in R} (f maps R to R is continuous) is a zero set? What would I be looking at to help determine that?

Are you talking about the empty set? An empty set is a set that has no valid elements and whose size is 0. It is also denoted by ∅. For example, {x∈R | x² < 0} = ∅.

Also, to add to this, {(x,f(x)) : x in R} = ∅ only if there's no function. Otherwise, we can find a valid coordinate point.

 

[Demon117]

Are you talking about the empty set? An empty set is a set that has no valid elements and whose size is 0. It is also denoted by ∅. For example, {x∈R | x² < 0} = ∅.

Also, to add to this, {(x,f(x)) : x in R} = ∅ only if there's no function. Otherwise, we can find a valid coordinate point.

No, I am not talking about an empty set. That is trivial.

 

[micromass]

Do you mean a set of measure zero?? A null set?

 

[gb7nash]

Well, if he's referring to this:

http://en.wikipedia.org/wiki/Zero_set

a zero set of a function f is the subset of R on which f(x) = 0. Basically, the set of roots of a function. To check that a given set is a zero set, just plug in each value into f and assert that f = 0. If that's not what he wants I'm not sure.