Mathematics - 2024-03-17 (page 2 of 2)

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Jakobian

21:00

and thats true

Pizza

Thanks!!! @Jakobian @leslietownes

I just ask one last thing

leslie townes

columbo!

Jakobian

one of my favourite shows

Pizza

why doesn't a symmetrical relation imply that it is also asymmetrical?

Jakobian

A relation $R$ is symmetric if $xRy$ implies $yRx$. Its asymmetric when $xRy$ implies that $\lnot yRx$

If there exists $x, y$ such that $xRy$ and $R$ is symmetric, then $R$ cannot be asymmetric, because that would imply that $yRx$ is not true. But its true, since its symmetric

That said, $R = \emptyset$ is an example of relation thats both symmetric and asymmetric

Pizza

21:19

asymmetric if: $xRy$ and $yRx$ $\Rightarrow x = y;$

Jakobian

I see, I was using the wikipedia definition

In this case, if $xRy$ and $R$ is both symmetric and asymmetric (anti-symmetric on wikipedia), then $yRx$ and so $x = y$

So if $R$ is both symmetric and anti-symmetric, then $R\subseteq \Delta_A$

and conversely, if $R\subseteq \Delta_A$ then $R$ is both symmetric and asymmetric

So examples of such relations exist, and there is more of them than using previous definition

one example is equality like above i.e. $xRy$ iff $x = y$

Pizza

Could you give me two examples (according to the definition I sent), one that imply both and of one which does not imply the other

pls

Jakobian

what do you mean?

by imply both/does not imply the other

Pizza

21:37

symmetric if: $xRy$ $\Rightarrow yRx;$ asymmetric if: $xRy$ and $yRx$ $\Rightarrow x = y;$

Anyway, sorry if I reply after a while, but I'm on the phone, not on the PC :(

Jakobian

I know that those are your definitions, its just not clear what properties do you expect the examples to have

Sine of the Time

Jakobian was asking clarification about your question about the two examples

Thorgott

I think you should come up with those examples yourself

as Jakobian has already explained, most interesting relations aren't both symmetric and antisymmetric

so if you can produce interesting examples of symmetric and anti-symmetric relations, you will also have plenty of counter-examples

Pizza

I didn't quite understand the difference between the two definitions

(the ones I sent 10 minutes ago)

symmetric if: $xRy$ $\Rightarrow yRx;$ asymmetric if: $xRy$ and $yRx$ $\Rightarrow x = y;$

Jakobian

lets say you have a relation on a set $A = \{1, 2, 3, ..., n\}$

Then you can describe a relation $R$ on $A$ in terms of a matrix having entries of $0$ and $1$

$B_{ij} = 1$ if $iRj$ and $B_{ij} = 0$ otherwise

Thorgott

21:49

they say different things, so they are different definitions

Jakobian

what does it mean for a relation $R$ on $A$ to be symmetric/asymmetric, in this example? In terms of the matrix $B = (B_{ij})_{i, j = 1}^n$

Thorgott

again, I suggest thinking about examples

consider something intuitive like a set $S$ consisting of the members of a family and let $R$ be the relation $xRy$ iff $x$ is an ancestor of $y$. is this symmetric? is it antisymmetric?

Jakobian

what does being an ancestor mean?

Sine of the Time

I think these are a bit advanced for Pizza

Thorgott

@Jakobian you know

Jakobian

21:52

I don't actually know what it means

oh you mean like a real life family

not a family of sets

Thorgott

yeah lol

Sine of the Time

@pizza try to find an example of a relation that is antysimmetric but it's not symmetric

Jakobian

@SineoftheTime I think people like to think in terms of matrices

Xander Henderson

@Thorgott The first time I read that, I read the last word as "antisemitic".

Sine of the Time

I don't know he's familiar with matrices, since he's at the beginning of a linear algebra course

Thorgott

21:57

@XanderHenderson oof, I prefer the "family of sets" misunderstanding

Jakobian

@XanderHenderson is it an antisemitic family? This will be on the test

Dannyu NDos

22:08

A: Second countability of compact open topology

To be concrete, here's my answer: If $X$ and $Y$ are second-countable and if $X$ is locally compact Hausdorff, $C(X,Y)$ is second-countable. Proof: Being second-countable locally compact Hausdorff, $X$ admits a countable basis whose every basic open set has compact closure; let $\mathcal{U}$ be o...

Any feedbacks?

Jakobian

@DannyuNDos X admits a countable basis whose every basic open set has compact closure

Pizza

Thanks everyone, I'll see better tomorrow, because I don't understand much on the phone

Jakobian

this is true, but can you justify it?

Pizza

I cant see the $$

Dannyu NDos

@Jakobian Edited.

Thorgott

22:14

besides that I'm not sure if this question really needed another answer, the argument seems incomplete

you're just showing a basic open in one contains a basic open in the other, but you need to show it is a union of such

Dannyu NDos

@Thorgott Munkres' Topology, p. 79.

Lemma 13.3.

Thorgott

I prefer communicating in words than with posting numbers from some book

I know what you're trying to do and I'm saying you didn't quite do it correctly

Jakobian

Yeah, you need to take $f\in \bigcap_{i=1}^n B_i$

and then show there is open $U$ in the other topology such that $f\in U\subseteq ...$

Thorgott

yeah, you need to show that every such element is contained in a basic open of $\mathcal{F}^{\prime}$

Jakobian

so what is missing is this taking of some arbitrary element

alternatively you can be lazy and cite Dugundji

Thorgott

22:21

well, the question already has an answer with a citation

Jakobian

no one actually gave any arguments so I think its fine

it was more of a discussion

Dannyu NDos

22:33

Edited.

Jakobian

23:00

@DannyuNDos you wrote that $V_{ij}$ is contained by $V_i$. What does that mean?

Dannyu NDos

23:23

Quite literally, $V_{i,j} \subset V_i$.

Jakobian

23:43

@DannyuNDos yeah this is wrong

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