# Cartesian plan - ordered pair

#### shahar

##### Full Member
How I can (if possible) that one point is only one order pair of the plane and one ordered pair is only one point on Cartesian plane?

#### Dr.Peterson

##### Elite Member
This is how we define the Cartesian plane, representing each point on a plane by the ordered pair (x,y) of its coordinates.

Why do you think it can't be possible?

#### shahar

##### Full Member
This is how we define the Cartesian plane, representing each point on a plane by the ordered pair (x,y) of its coordinates.

Why do you think it can't be possible?
How we define every point by unique place? How can we show that the place of the point is unique?

#### Dr.Peterson

##### Elite Member
How we define every point by unique place? How can we show that the place of the point is unique?
A "point" and a "place" are the same thing. Where does the ordered pair come into your question?

The coordinate plane concept is based on choosing two lines in the plane (conventionally perpendicular, though that was originally not assumed), and projecting any given point onto each line. This results in two coordinates, called x and y. Clearly the ordered pair is unique; and the construction clearly can be reversed to obtain a unique point for any given ordered pair.

But again, what is the reason for your question? What observation caused you to question whether the point is unique?

All you are doing so far is asking questions, like a two-year old! They don't reveal what you are actually thinking, so I can't tell what kind of answer is needed to satisfy you.

#### lev888

##### Senior Member
How we define every point by unique place? How can we show that the place of the point is unique?
An ordered pair of coordinates defines a unique point in a particular coordinate plane because 2 perpendicular lines have one intersection.

#### shahar

##### Full Member
A "point" and a "place" are the same thing. Where does the ordered pair come into your question?

The coordinate plane concept is based on choosing two lines in the plane (conventionally perpendicular, though that was originally not assumed), and projecting any given point onto each line. This results in two coordinates, called x and y. Clearly the ordered pair is unique; and the construction clearly can be reversed to obtain a unique point for any given ordered pair.

But again, what is the reason for your question? What observation caused you to question whether the point is unique?

All you are doing so far is asking questions, like a two-year old! They don't reveal what you are actually thinking, so I can't tell what kind of answer is needed to satisfy you.
Only I was thinking, that it can be proven. Why I cannot prove it?

#### lev888

##### Senior Member
Only I was thinking, that it can be proven. Why I cannot prove it?
You can. Draw a coordinate plane. Pick any point. Its x coordinate is found by projecting the point onto the x axis. This method produces a unique result - 2 lines have one intersection. Same for y coordinate.
Now pick any (x,y) pair. It corresponds to a unique point because again, 2 lines have one intersection.

#### Dr.Peterson

##### Elite Member
Only I was thinking, that it can be proven. Why I cannot prove it?

You're right that it can be, and has been, proved. You've been given two suggestions how to do so. Now you have to tell us what is lacking in you, that prevents you from carrying it out.

It could be that you are not thinking in the context of specific axioms and theorems, so you have no foundation on which to build a proof; or it could be that you have not practiced writing proofs, so you have no tools for connecting the facts you know to this one. Or it may be that you don't have a specific definition of the Cartesian plane, so you don't have a clear goal toward which to direct a proof.

In any case, tell us what you tried, and in what way you failed to prove it.

#### shahar

##### Full Member

You're right that it can be, and has been, proved. You've been given two suggestions how to do so. Now you have to tell us what is lacking in you, that prevents you from carrying it out.

It could be that you are not thinking in the context of specific axioms and theorems, so you have no foundation on which to build a proof; or it could be that you have not practiced writing proofs, so you have no tools for connecting the facts you know to this one. Or it may be that you don't have a specific definition of the Cartesian plane, so you don't have a clear goal toward which to direct a proof.

In any case, tell us what you tried, and in what way you failed to prove it.
O.K.
The Foundation that I find is by definition of relation R in Set Theory.
The relation is ordered pairs that have a relation,
How can I use relation in Set Theory to Define that every point in the plane is unique and note by ordered pair and ordered pair is unique in the Cartesian plane?
I THINK BY DEFINITION THE RELATION I CAN PROVE IT. CAN I?

#### Dr.Peterson

##### Elite Member
O.K.
The Foundation that I find is by definition of relation R in Set Theory.
The relation is ordered pairs that have a relation,
How can I use relation in Set Theory to Define that every point in the plane is unique and note by ordered pair and ordered pair is unique in the Cartesian plane?
I THINK BY DEFINITION THE RELATION I CAN PROVE IT. CAN I?
I don't know. Can you?

But I don't see how what you say is relevant. What specific relation do you have in mind?

#### shahar

##### Full Member
I don't know. Can you?

But I don't see how what you say is relevant. What specific relation do you have in mind?
{{1, 2}} != {{2, 1}}
If I show that set {1, 2} not equal to {2, 1} by relation of =. I can show that every point is unique?
The question is how I begin and go from point (2, 1) to set {{2, 1} and etc.
And what other relation is there between point (2, 1) to point (1, 2)?
Something like this.
You have an idea how to translate the point to set like that?...

#### Dr.Peterson

##### Elite Member
{{1, 2}} != {{2, 1}}
If I show that set {1, 2} not equal to {2, 1} by relation of =. I can show that every point is unique?
The question is how I begin and go from point (2, 1) to set {{2, 1} and etc.
And what other relation is there between point (2, 1) to point (1, 2)?
Something like this.
You have an idea how to translate the point to set like that?...
Are you talking about sets, or about ordered pairs? Why bring in the former? And what does your notation {{1, 2}} mean?

The set {1, 2} is the same set as {2, 1}; the ordered pair (1, 2) is not the same ordered pair as (2, 1). That is a large part of the reason for the one-to-one correspondence of ordered pairs to points.

There is no reason to relate the pairs or points (1, 2) and (2, 1).

#### pka

##### Elite Member
The Foundation that I find is by definition of relation R in Set Theory.
The relation is ordered pairs that have a relation,
I THINK BY DEFINITION THE RELATION I CAN PROVE IT. CAN I?
To: Shahar, I think that if you would improve your understanding of English vocabulary.
Now please tell me what what textbook is set theory you studied any the university(college) were you did it.

#### shahar

##### Full Member
I meant that the set that contains another set different from the set that contain the opposite set. No?

#### Dr.Peterson

##### Elite Member
I meant that the set that contains another set different from the set that contain the opposite set. No?
Does that mean anything??