How to Formulate Sets and Equivalence Classes in Set Theory Notation?

[jaejoon89]

How do you write in proper set theoretic notation that
a set A = (x,x) where x is a non-negative real number?

Also, (x_1, y_1) R (x_2, y_2) if x_1 ^2 + y_1 ^2 = x_2 ^2 + y_2 ^2
The equiv. classes are circles at (0,0), right?
How do you write this formally (using set theoretic notation)?

 

[foxjwill]

If it's clear from the context that you're working in the real numbers with the standard ordering, then you'd probably write $$A=\{(x,x)|x>0\}$$. Otherwise, you'd usually write either $$A=\{(x,x)|x\in\mathbb{R}^+_0\}$$ or $$A=\{(x,x)|x\in[0,\infty)\}$$, although these are by no means the only conventions.

In general, to define a set using set notation, you need to specify (a) a collection of variables, (b) what condition those variables must satisfy, and (c) how the variables are combined to make an object in the overall set. In other words, a set S is given in set notation by

$$S=\{L(x_1,x_2,\ldots)|Q(x_1,x_2,\ldots)\}$$​

where Q is some condition (e.g. the condition that $$x_1$$ is a blue ball, $$x_2$$ is a real number, and all the other variables are stars in the Andromeda galaxy) and L specifies a way of combining variables.

 

[foxjwill]

Also, (x_1, y_1) R (x_2, y_2) if x_1 ^2 + y_1 ^2 = x_2 ^2 + y_2 ^2
The equiv. classes are circles at (0,0), right?
How do you write this formally (using set theoretic notation)?

Yes, they are. Formally, the equivalence class [x,y] of (x,y) is

$$[x,y]=\left\{(x',y')|x'^2+y'^2=x^2+y^2\right\}.$$​