What is a Null Set and Why Does It Have Measure Zero?

[mooberrymarz]

Wot exactly is a null set? i don't understand it. if anyone could explain that wuld be wikid.

 

[matt grime]

it would depend on the context, but it is, generically, something that is zero.

 

[vadik]

It exists to make difference between "something" and "nothing". A set that contains something is like something and the null set is nothing (contains nothing)

 

[matt grime]

No, that is not necessarily true. A null set, could be, as I suspect it will be in this case, a set of measure zero.

 

[vadik]

ok..maybe my english is not good enough.. what is that called: ø?

 

[vadik]

isn't it a null set? or an empy set?

 

[matt grime]

that is the empty set. It is a null set in the sense that its cardinality is zero.

 

[vadik]

is there any other set, whiches cardinality is zero exept the empty set?
I don't think so

 

[matt grime]

No, it, the empty set, is unique. But that doesnt' have any bearing on what a null set is until we see what situation we are dealing with.

 

[mooberrymarz]

For example.. the set of rationals and irrationals. rationals are a null set and wot bout irrationals?? Matt could u explain in dummy maths why rationals are a null set?

 

[matt grime]

As I thought, a null set is one that has measure zero. Example: and countable subset of R (Eg the rationals): let x_i be an enumeration of the set, round each point x_i consider the interval e/2^i, then the measure of the set is less than the sum over i of e^2^i = e. e was arbitrary hence it has measure zero.