Real No.s: Writing in Set Builder Form

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In summary: For example, the set of natural numbers can be written as N = {1,2,3,4,...}. However, that is not a set in and of itself. You would need to know what the pattern is in order to be able to say that x belongs to the set.
  • #1
johncena
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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 ?
 
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  • #2
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.
 
  • #3
Do you mean you want to pair off a rational with an irrational, i.e., define a bijection between them?
 
  • #4
CRGreathouse said:
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 [tex]\in[/tex]Q or x[tex]\in[/tex]T}
Can't we write R in roster form ?
 
  • #5
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.
 

1. What is the set builder notation for real numbers?

The set builder notation for real numbers is {x | x is a real number}.

2. How do you write real numbers in set builder form?

To write real numbers in set builder form, you use the set builder notation {x | x is a real number}. This means that x can be any real number.

3. What is the difference between set builder notation and interval notation for real numbers?

The main difference between set builder notation and interval notation for real numbers is the way the numbers are represented. Set builder notation uses the set notation of curly braces and a variable to represent the set of numbers, while interval notation uses parentheses or brackets to represent a range of numbers.

4. How do you represent infinite and finite sets of real numbers in set builder form?

Infinite sets of real numbers can be represented in set builder form by using the ellipsis symbol (...). For example, the set of all real numbers greater than 5 can be written as {x | x > 5} or {x | x > 5, x is a real number}. Finite sets of real numbers can be written by listing out the individual elements within the curly braces. For example, the set {1, 2, 3, 4} can be written as {x | x = 1, 2, 3, 4}.

5. What are some common applications of set builder notation for real numbers?

Set builder notation for real numbers is commonly used in mathematics and computer science. It is used to represent sets of real numbers in mathematical equations and expressions, as well as in programming languages to define sets of data. It is also used in statistics and probability to represent sample spaces and event sets.

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