How to determine a equivalence relation?

Question

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?

That last equality sign in your third line is thrilling, @Googi...What did you really mean?? — DonAntonio, Mar 25 '14 at 17:35
sorry...thats basically the answer that ist system of representatives = $ \lbrace{(0,x)|x \in \mathbb{R}\rbrace}$ — Googme, Mar 25 '14 at 17:44
As far as I can tell the the equivalence classes are just horizontal lines in the plane. — JP McCarthy, Mar 25 '14 at 17:50
Well, don't write it thay way as what you wrote means that the cartesian product $;\Bbb R^2\times\Bbb R^2;$ equals that, whic of course is absurd... — DonAntonio, Mar 25 '14 at 17:50
@JpMcCarthy, indeed...and that's what Googi wrote after explaining his notation. — DonAntonio, Mar 25 '14 at 17:51

score 0 · Answer 1 · answered Mar 25 '14 at 17:51

0

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?

answered Mar 25 '14 at 17:51

Michael Hoppe

18,103
3
32
49

so in the present case the tuple (1,3) or (1,x) is also possible? – Googme Mar 25 '14 at 17:55
What do you mean by that, @Googi ? And why haven't you corrected your post, yet? – DonAntonio Mar 25 '14 at 18:08
@Googi No, your tuples don't share the same second coordinate. – Michael Hoppe Mar 26 '14 at 14:35

How to determine a equivalence relation?

1 Answers1