Sets and ordered pairs: Kuratowski set definition of an ordered pair

[swgeek]

Hi;
I am beginning the self study of mathematics in order to regain some of the knowledge of my youth (former engineering student through differential equations), and am attempting to do it on a more rigorous level than before. I am reading Zakon's Basic Concepts of Mathematics and concurrently teaching my self Latex by taking notes in that markup language (rather slow slogging, but I am trying).
I understand the basics of Cartesian Coordinates, and concepts of an ordered pair, but have come to the concept that if Sets are a fundamental concept of mathematics, then anything can be expressed within the context of sets. I have found the following Kuratowski set definition of and ordered pair:
(a,b) := {{a},{a,b}}
Now I understand a set with the member a, and a set with the members a and b, but I am unsure how to read that, and how it describes an ordered pair, or Cartesian Coordinate. I would read the right side of that as "The set of sets {a} and {a,b}". How does that relate to an ordered pair, or a point in two dimensional space?
Thanks
swgeek

 

[Dr.Peterson]

Hi;
I am beginning the self study of mathematics in order to regain some of the knowledge of my youth (former engineering student through differential equations), and am attempting to do it on a more rigorous level than before. I am reading Zakon's Basic Concepts of Mathematics and concurrently teaching my self Latex by taking notes in that markup language (rather slow slogging, but I am trying).
I understand the basics of Cartesian Coordinates, and concepts of an ordered pair, but have come to the concept that if Sets are a fundamental concept of mathematics, then anything can be expressed within the context of sets. I have found the following Kuratowski set definition of and ordered pair:
(a,b) := {{a},{a,b}}
Now I understand a set with the member a, and a set with the members a and b, but I am unsure how to read that, and how it describes an ordered pair, or Cartesian Coordinate. I would read the right side of that as "The set of sets {a} and {a,b}". How does that relate to an ordered pair, or a point in two dimensional space?
Thanks
swgeek

The basic idea is that a set has no order to it, so in order to represent an ordered pair purely in terms of sets, you have to somehow distinguish the first element from the second. What he has done is to distinguish the first element by putting it in a set by itself, and then identify the second element by putting it in a set together with the first.

So, for example, if I gave you the set {{5,3},{3}}, you would find the singleton set {3}, which tells you that the first element is 3; then find the set with two elements, {5,3}, and remove 3 from it, to see that the second element is 5. The ordered pair is (3,5). Note that I deliberately put everything out of order in this example, to emphasize that order is irrelevant.

What ordered pair is represented by {{3,3},{3}}?