Real No.s: Writing in Set Builder Form

[johncena]

Why can't we write the set of real no.s in roster form?
In set builder form, R = {x:x 'belongs to' T or x 'belongs to' Q}
so , if we write one rational no.& one irrational no in a set., that will be the set of real no.s isn't it ?

 

[CRGreathouse]

If I understand you correctly, you're asking why a list of real numbers can't be composed of a list of rational numbers and a list of irrational numbers. The answer, if that was your question, is that the irrational numbers are too large to list.

 

[honestrosewater]

Do you mean you want to pair off a rational with an irrational, i.e., define a bijection between them?

 

[johncena]

If I understand you correctly, you're asking why a list of real numbers can't be composed of a list of rational numbers and a list of irrational numbers. The answer, if that was your question, is that the irrational numbers are too large to list.

I don't wan't to create a list , i wan't a set...in roster form...
for eg:- the set of natural no.s N = {1,2,3,4,...}
the set of whole no.s W = {0,1,2,3,...}
I know that real no.s cannot be listed in order...but since there isn't any importance of order in sets, that is {W,O,L,F} = {F,O,L,W},I think we can write the set of real no.s in roster form by listing one or two elements in it and then putting dots...
like this , {pi,root 2,1,5/2...}
But my teacher said that we can't write the set of real no.s in roster or tabular meathod since it includes no.s of different patterns ...
But since it can be written in set builder form like this ,
R = {x:x $$\in$$Q or x$$\in$$T}
Can't we write R in roster form ?

 

[honestrosewater]

The problem with that idea is that it does not define one set. There is no way to know what the pattern is, i.e., what the "..." is supposed to stand for. The set builder notation tells you exactly which numbers are in the set, and in order for some object to be a set, you need to be able to tell, for every object x, whether or not x is a member of the set.

The other roster definitions that you mention are not clear enough in themselves either. They are a shorthand for referring to a set that is already familiar.