Is the Definition of an Ordered Pair Set Theoretic?

[reb659]

Homework Statement

Determine whether or not the following definition of an ordered pair is set theoretic (i.e. you can distinguish between the "first" element and the "second" element using only set theory).
(x,y)={x,{y}}

Homework Equations

The Attempt at a Solution

I am inclined to think no, but can anyone explain the rationale for why it does not work?

 

[Dick]

x is an element of {x,{y}}. Is y an element of {x,{y}}?

 

[reb659]

No. But why does it matter if it is in it or not?

 

[Dick]

No. But why does it matter if it is in it or not?

You asked whether you can distinguish between x and y using set theory. Being 'an element of a set' is set theory, isn't it?

 

[Hurkyl]

You asked whether you can distinguish between x and y using set theory.

Here is an ordered pair, by this definition.

{{2}, {1}}​

Can you tell me what the left element is, and what the right element is?

 

[Dick]

Here is an ordered pair, by this definition.

{{2}, {1}}​

Can you tell me what the left element is, and what the right element is?

The definition of (x,y) is {x,{y}}, isn't it? Not {{x},{y}}.

 

[Hurkyl]

The definition of (x,y) is {x,{y}}, isn't it? Not {{x},{y}}.

Remember that we're doing set theory, so x and y are variables denoting sets. There exists x and y such that (x,y)={{2},{1}}.

(Also, my 1 and 2 are meant to be naming sets whose precise identity isn't relevant. The typical naming scheme here is 0={}, 1={0}, 2={0,1}, ... Mainly, I just didn't want to have millions of braces lying around in my example)

 

[Dick]

Remember that we're doing set theory, so x and y are variables denoting sets. There exists x and y such that (x,y)={{2},{1}}.

(Also, my 1 and 2 are meant to be naming sets whose precise identity isn't relevant. The typical naming scheme here is 0={}, 1={0}, 2={0,1}, ... Mainly, I just didn't want to have millions of braces lying around in my example)

Ok, I see what you are saying. {{2},{1}} corresponds to either the ordered pair ({2},1) or ({1},2). If I know x and y, I can tell you which is first. But if I don't know what x and y are to begin with, I can't tell you what they are from {{2},{1}}. Thanks.