Help me

[cinder yoke]

What part do you not understand?

An equivalence relation is a relation $\sim:X\times X\to X$ which has $\forall x,y,z\in X$ \begin{enumerate}
\item $x\sim x$
\item $x\sim y\implies y\sim x$
\item $x\sim y$ and $y\sim z\implies x\sim z$
\end{enumerate}
Then you can say the set of elements which are equivalent to $x$ (denoted $[x]_{\sim}$) is called the equivalence class of $x$. Note that the choosing of $x$ as the representative was arbitrary.

[nocturne karma]

SelahW

[cinder yoke]

For example we can take $X=\mathbb{Z}$, and say $x\sim y$ if $x-y\mid 10$. Then there are 10 equivalence classes, e.g. [[3]={\dots,-17,-7,3,13,23,\dots}] or [[6]={\dots, -14,-4,6,16,26,\dots}]

[nocturne karma]

SelahW

[cinder yoke]

@vernal rock

[vernal rock]

what does Idea mean?

[vernal rock]

@cinder yoke

[cinder yoke]

When you have a set and an equivalence relation defined on the set, you can mod out by the relation to get a new set. It's called the quotient set and it's elements are equivalence classes

This usually comes up in terms of some algebraic structure (group ring module etc) because normal subgroups, ideals, and submodules define a unique equivalence relation on their resp. structures

[vernal rock]

I just dont understand thing because i dont think like nlab

[cinder yoke]

Yes it tries to be general, that's kind of the point

[vernal rock]

as i see they starting with quotient set not from its elements

and define classes as elements of set not vice versa

maybe i dont get things right

[cinder yoke]

Well you start with the set X, and an equivalence relation $\sim$ on X. With that you get a new set $X/\sim$

[nocturne karma]

SelahW

[cinder yoke]

That's the quotient set

Everything that was simply equivalent in X is now equal in $X/\sim$

[nocturne karma]

SelahW

[vernal rock]

like they say:

• X/~ exists
• [x]~ = [y]~ iff x ~ y

[cinder yoke]

Yes

[vernal rock]

(also f: S -> S/~)

[cinder yoke]

That's this

[cinder yoke]

This is usually denoted with pi and I've always heard it called "the natural reduction map"

It sends each element to its equivalence class

With the example I gave above, we have our set $X=\mathbb{Z}$. The equivalence relation is $x\sim y\iff x-y\mid10$.

[nocturne karma]

SelahW

[cinder yoke]

Then we can form the quotient, and we get a new set $\mathbb{Z}/\sim$ where there are 10 elements, and elements are equal if they were equivalent in the other set.

[nocturne karma]

SelahW

[cinder yoke]

In the set $\mathbb{Z}$, 16 and 6 are equivalent because $16-6=10\mid10$. So the equivalence classes $[16]$ and $[6]$ are equal in the quotient.

[nocturne karma]

SelahW

[cinder yoke]

Again the cat lab definition is not the one I would choose if I were an undergrad or explaining it to an undergrad

Probably the way this should be learned is through group theory. Nobody generally just quotients a set lol, it is usually a quotient group, ring, or module

[vernal rock]

there set

[vernal rock]

Help me

.solved