geometric descriptions real number line

[agile trout]

I don’t understand what to do

Describe geometrically the following sets on the real number line

I get x is bigger than a and smaller or equal to b

now what tho

[pulsar ivy]

Okay, so look at the inequalities here

Think about what that tells us about the end behavior on the set

[agile trout]

End behavior?

[pulsar ivy]

For the x <= b part, note that this set includes b but no numbers greater than b

[pulsar ivy]

Just the way the interval behaves at the the two ends

[agile trout]

oh

[agile trout]

yes and it doesn’t include a

[pulsar ivy]

Whereas for a < x, a is not included in the set

Do you know open and closed intervals?

[agile trout]

not really

(a, b]

x of (a,b]

a <= x <= b

is [a,b] = closed

a < x < b

is (a,b) = open

[pulsar ivy]

yes

[agile trout]

smh thought no pre requisites to this book

[pulsar ivy]

basically all you need to do is treat the set as shaped like (a,b]

So it is open on the a-side and closed on the b-side

[agile trout]

How should the answer look like tho

x € (a,b]?

[pulsar ivy]

Yes

Well it wants a geometric description

I think “interval from a to b with a excluded and b included” is sufficient

[agile trout]

I see

What about |x|<0

the standard -a < x < a don’t work here

[pulsar ivy]

|x| < 0?

[agile trout]

Yeah

[pulsar ivy]

Ah, it’s empty

It’s just no points at all

[agile trout]

Empty set

[pulsar ivy]

Yes

[agile trout]

alr alr I get it

so it’d also be true that |x| >= 0 right

always

[pulsar ivy]

yes

[agile trout]

a)
b) true
c) true
d)
e)
f) true

How about this

i don’t get this I’m just seeing what makes sense

{a} obviously is part of (a,b) according to the definition and so is (f)

c makes sense cuz that’s just {{a}, {a,a}}

[pulsar ivy]

Yes, you’re right for all of them

[agile trout]

what’s the math here I don’t get it

[pulsar ivy]

It’s axiomatic set theory

[agile trout]

the author maybe mentioned axiom like once like uh where is that explanation

[pulsar ivy]

I think you understand it though since you correctly identified which were correct

[agile trout]

Yeah but like if you were to ask me

What’s the difference between {a,b} and {{a}, {a,b}}

if you were to describe both on like a real number line

[pulsar ivy]

Well, I don’t know if one can really describe the latter on a real line

Since it is not a set of real numbers

It’s a higher-order set (it contains sets as its elements)

[agile trout]

Okay

It feels like I learnt nothing by doing these exercises though😭😭

and

then this is so random like

Oh I have to use the def

For a ≠ b
(a,b)
{{a}, {a,b}}
{{u}, {u, v}}
(u,v)

like what am I supposed to do after using the definition😭

[pulsar ivy]

I mean the definition is basically ass

It shouldn’t be used for anything ever

I have to go now

[agile trout]

how do you even do this proof like it seems so useless

[agile trout]

pce

Yo wawi so

(ii)

[elfin stag]

Ah

[agile trout]

I really don’t get the use of this or even how to go about it

[elfin stag]

Do you know sets?

[agile trout]

Yeah

Like I can prove and understand De Morgan’s Laws with existential and universal quantifiers

[elfin stag]

Oh then you’re good lol

[agile trout]

but this exercise seems so random like wtf

[elfin stag]

Ok now from the def we have $(a,b) =\qty{\qty{a},\qty{a,b}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

It’s actually basic set theory

[agile trout]

okay

[elfin stag]

Ok first question

Is $a\in(a,b)$

[inner arrow]

Wawi #NwoWifer

[agile trout]

I understand (i)

[elfin stag]

Use this definition

[agile trout]

only b c and f are true

it’s about (ii)

[elfin stag]

It’s only b and f lol

[elfin stag]

Ah

[agile trout]

oh magma checked my answers and I had c as true

and she said all were right

[elfin stag]

$(a,a) =\qty{\qty{a}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

You’d get that

Is that equal to {a} though?

[agile trout]

yeah so if (a,a) then {{a, {a,a}}

[agile trout]

And that’s just {{a}}

[elfin stag]

But {{a}} isn’t {a} right?

[agile trout]

uh well {a} is a set and {{a}} is two

yes

[elfin stag]

They aren’t equal right?

So even c is false

[agile trout]

I see

[elfin stag]

Alr now let’s do Q2

[agile trout]

prove (a,b) = (u,v) if and only if a = u and b = v

[elfin stag]

$(a,b) = \qty{\qty{a},\qty{a,b}}$

$(u,v) = \qty{\qty{u},\qty{u,v}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

Right?

[agile trout]

Yes this I get now what

[elfin stag]

Now replace every u with a and v with b

So you get

$(u,v) =\qty{\qty{a},\qty{a,b}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

Then you get $(u,v)=(a,b)$ from the definition of $(a,b)$

[inner arrow]

Wawi #NwoWifer

[agile trout]

Oh you have to phrase it like that

But then it also says

To do it when a = b and when a ≠ b

[elfin stag]

The case division, ik

[elfin stag]

Alr let’s do it for a=b then

That implies that u=v as well, right?

[agile trout]

Ye

a=u=b=v

[elfin stag]

So we get if $a=b$

$(a,b)=\qty{\qty{a}}$

$(u,v)=\qty{\qty{u}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

Now as $u=a$

$(u,v) =\qty{\qty{a}}$

[inner arrow]

Wawi #NwoWifer

[elfin stag]

And from there it is crystal clear

[agile trout]

I understand all steps but like wtf is the use of this exercise

what do we learn here

we already had the definition so it’s just changing the elements to rewrite it

[elfin stag]

Nothing

They literally just messed with yoy

*you

[agile trout]

😔

like bro

It’s so fucking random, exercise 4 was to prove de Morgan’s laws

And then exercise 5

is literally closed and open intervals

[elfin stag]

Wow

Bro you need a better book

[agile trout]

some guy recommended me this I think it was Coffey or something

[elfin stag]

💀

He recommended you this book-

Lmfao

[agile trout]

Is it bad 😭💀

[elfin stag]

No it’s rigorous set theory, so its good
But the exercises are too easy

[agile trout]

Yeah I don’t feel like I’m learning anything

by doing these exercises

[elfin stag]

Lmfao

[elfin stag]

Ok, how about this

[agile trout]

me proving -(-A) = A

like bro

[elfin stag]

If the answer is like too obvious

Then skip the question

[agile trout]

It’s Elias Zakon’s Anal 1

[elfin stag]

Or yk, i could give you another book on set theory
But uhhh it’s not exactly like the one you see here

[elfin stag]

Ah

His exercises are too easy

I’m studying from Rudin

The exercises force you to think in that

[agile trout]

the theory is nice but yk I’d like some good exercises too

[elfin stag]

Ok do you know what a countable set is?

[agile trout]

I see some harder ones tho coming

[elfin stag]

Ooh alr

[elfin stag]

Ah those are ok exercises

[agile trout]

Cartesian product tryna ruin everything

[elfin stag]

Hahahahaha

[elfin stag]

Do yk what this means though?

[agile trout]

well I’m guessing a non infinite set excluding empty set

idk tho

[elfin stag]

Ah nvm

[agile trout]

What’s the material called I need to learn to answer these

[elfin stag]

Ah these are funny

[agile trout]

how do I describe like what’s the answer format

[elfin stag]

Easy as well

[elfin stag]

www.desmos.com

[agile trout]

oh looks neat

[elfin stag]

Wait no that’s cheating lmfao

[agile trout]

Oh

[elfin stag]

Tbh for this just draw the plane in your mind

And then plot it visually

(Then check your answer with desmos)

For example,

For (i) the answer is the entire region below the line x=y but not including the line

[agile trout]

Ohh

But why does this guy hit me with this when it’s about just set logic not this boring stuff

[elfin stag]

Idk lol

[agile trout]

Do you have a good set problems pdf

[elfin stag]

I don’t uhhh

[agile trout]

If I search set problems I get stuff like what’s the intersection of A = {1,2,3} and B = {3,4,5}

🤯🤯

[summer rock]

@agile trout this is just basic hs math though

If you paid attention in hs you'll be fine

(the prereq.)

[agile trout]

I’m not used to the terminologies for a lot, cuz I think my country’s syllabus is quite different to the standard in USA or UK

[summer rock]

that's possible

[agile trout]

like idk what describing geometrically means in this sense

[summer rock]

it means draw the set

and cover the inside with something

[agile trout]

cover the inside?

oh the set

[summer rock]

yes

like a handwavy colouring