I have a problem to understand the following output:
Determine "representative system" or a "system of representatives" :).....for the following equivalence relation $R:=\lbrace{(x_1,y_1),(x_2,y_2)|y_1=y_2\rbrace}\subseteq \mathbb{R}^2 \times \mathbb{R}^2$
With $ \lbrace{(0,x)|x \in \mathbb{R}\rbrace}$ as the "representative system"
Could someone please provide step to step explanation?
The concept of equivalence relation is to consider two objects to be the same provided they're sharing a common property. By abstracting -- that means in the literal sense overlooking -- that property we're able to construct new objects. (According to my knowledge this is due to H. Weyl.)
In our case we consider two points to be equal iff their second coordinates are the same. In that sense $(4,3)$ equals $(-2311,3)$ equals any point whose second coordinate is $3$. So all points equal to $(4,3)$ form a new object, namely the parallel line to the first coordinate axis through $(4,3)$. Or through $(0,3)$.
Can you take it from here?