help-10

[thin star]

and for the potential energy due to gravity?

[spark stump]

Idk that

[thin star]

mgh

h is height from the reference point

does that sound familiar?

[spark stump]

Yes

[thin star]

so you have PE + KE at the top

[spark stump]

So

[thin star]

and PE + KE at the rollercoaster loop

just equate those

[spark stump]

Hm ok

Pe

Mgh

[thin star]

Okay let's do this. What's the kinetic energy at point 1

[spark stump]

50 * 9.8 *4+1/2 (50 * 0)^2

[thin star]

also g = 9.81 (approximately)

At any rate, what is the kinetic energy at point 1?

[spark stump]

Uh idk

[thin star]

Bruh you can't just say idk during the exam

[spark stump]

Yes

[thin star]

Try to think right now, that's why I'm guiding you

[spark stump]

But I actually dk

[thin star]

What's the formula

[spark stump]

Ok

Ke is 1/2mv^2

[thin star]

Ok good. What's v at point 1

[spark stump]

1=1/2mv^2

Hm

It is 0

[thin star]

Good! So that means there's 0 kinetic energy at point 1

It's all potential energy

[spark stump]

Ok

Hm oke

Kinetic energy is stored energy right

Preserved energy

And potential energy is

?

What's potentionalbagain

[thin star]

I wouldn't call it that, but the usual "elementary" definition is 'capacity to do work'

gravitational potential is mass x g x height from some refernce point

You need to establish where the height is 0

where do you think that should be

[spark stump]

Hm

The where h is 0

[thin star]

There's no one right answer here, I just want you to be consistent

[spark stump]

At point 3 obv

[thin star]

So the ground

[spark stump]

Yes

[thin star]

Okay good

we say h = 0 at the ground

so the potential energy is what?

at point 1

[spark stump]

potential energy is mgh

So

50 * 9.8*4

[thin star]

Good. keep that in mind for later

Now

[spark stump]

Which is

[thin star]

Let's look at the potential anergy at point 2

[spark stump]

196

Ok

[thin star]

what is it

[spark stump]

50 *3 *9.8

[thin star]

I know what you mean. The asterisk screws it up

[spark stump]

Yes

[thin star]

What about v?

at 3

[spark stump]

Velocity

What's the formula for that

[thin star]

oh sorry i meant at point 2

What's the formula for kinetic energy

[spark stump]

What's the formula

1/2mv^2

It will be

1/2 * 50 * v

[thin star]

Okay. So according to the conservation of energy: $m g h_1 + 0 = m g h_2 + \frac{1}{2}mv^2$

[warm shale]

TooManyCooks

[thin star]

the 0 on the left is the kinetic energy

at point 1

Do you understand where this came from?

[spark stump]

Hm

No

[thin star]

Okay

All I did is say Kinetic + potential energy at point 1 = kinetic + potential energy at point 2

That's why it's called "conservation of energy"

[spark stump]

Ok I see

Does it always have to mgh+0

[thin star]

Not always

But

[spark stump]

For it to be mgh+1/2mv^2

[thin star]

it would be VERY convenient if you can find places where one of them is 0

for example, at the ground, mgh = 0

[spark stump]

Hm?

[thin star]

or, when it starts from rest, kinetic energy = 0

[spark stump]

Ok

What's the 0 suppose to be

In terms of formula

[thin star]

mgh = 0 because h = 0 at the ground

But that's just an example

You can always adjust the height so that one of the energies is 0. Don't worry about that now though

For now let's focus on finding the answer to your question

[spark stump]

So it's not

Uh ok

[thin star]

Oh, what's your question

[spark stump]

Should I memorize that mgh+0=mgh +1/2mv^2

[thin star]

No

never do that

[spark stump]

Oh

[thin star]

What you should memorize is formula for kinetic energy and the formula for potential energy due to gravity

mgh

and 1/2 m v^2

[spark stump]

1/2mv^2

And mgh

[thin star]

Yes

[spark stump]

Ye that's ok formula sheet

Oke

On

[thin star]

conservation of energy is just adding those two

[spark stump]

Oh

That's on the lhs

[thin star]

Yes

[spark stump]

What about the hrs

Rhs

Same thing

?

[thin star]

Yes, but with different height and velocity

$m g h_1 + \frac{1}{2}m v_{1}^{2} = m g h_2+ \frac{1}{2}m v_{2}^{2}$

[warm shale]

TooManyCooks

[thin star]

h1 and v1 for LHS

[spark stump]

Oh ok

[thin star]

h2 v2 for RHS

[spark stump]

Ez

[thin star]

It just so happen that in your problem, v1 = 0

[spark stump]

So just same thing but different points on each side

[thin star]

Yes, but the velocity and height may be different

[spark stump]

Yea

[thin star]

Anyway, back to your problem

[thin star]

You have $m g h_1 + 0 = m g h_2 + \frac{1}{2}mv^2$

[warm shale]

TooManyCooks

[thin star]

Can you isolate v?

[spark stump]

Well he

Ye

We know the mgh too

Mgh 2 is

50 * 9.8*3

Mgh1 is 50 * 9.8*4

So the we have

Uh

I know how to isolate v

But for now I'm going to use calculator

,w 509.84=509.83+1/2(50x)^2

[warm shale]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by fwgx

[spark stump]

Uh

[thin star]

You know you could also just analytically solve for v

[spark stump]

Yea

[thin star]

$v = \sqrt{2g(h_1-h_2)}$

[warm shale]

TooManyCooks

[spark stump]

That's the rearranged equation?

[thin star]

Yes

From this

$m g h_1 + 0 = m g h_2 + \frac{1}{2}mv^2$

[spark stump]

I see

[warm shale]

TooManyCooks

[thin star]

m cancels

[spark stump]

Lemme try that

Wtf

mgh1-mgh2=1/2mv^2

Sqrt((mgh-mgh)/(1/2m))=v

Uh

It's whatever

So ww just plug values now?

We

[thin star]

Yes

$v = \sqrt{2g(h_1-h_2)}$

[warm shale]

TooManyCooks

[spark stump]

Sqrt(19.6)

Which is

4.43

So v is 4.43?

[thin star]

sounds right. 4^2 = 16, pretty close

[spark stump]

Are you sure

[thin star]

Why?

[spark stump]

This seems to be a lot of work for a one mark question

[thin star]

That's not my fault. Ask your instructor why that is

[spark stump]

Xd

[thin star]

Although you're right

[spark stump]

Can we move on to next q

[thin star]

This is a bit of work for 1 point

sure

[spark stump]

I can ask more than 1q per thread?

[thin star]

sure

[spark stump]

That's okay?

Feel like this is easy just idk which formula to plug in

[thin star]

Okay

So what do you think?

Have you even tried this?

[spark stump]

Nope

[thin star]

Okay

[spark stump]

I got stuck trying it

Yesterday

[thin star]

Sketch first

[spark stump]

I did prac exam yesterday

[thin star]

When in doubt, draw it

[spark stump]

What's the 50hz

Like amplitude?

[thin star]

it's the frequency of oscillation

[spark stump]

Hm

[thin star]

or i guess in this case, the vibration

[spark stump]

Is that like

Period or amplitude

[thin star]

It's related to the period

not the amplitude

[spark stump]

I see

[thin star]

Sorry to leave you hanging but I actually have to leave in like 5 minutes lol I didn't expect you to have a follow up question

If you still need help, feel free to ping the helpers

[spark stump]

V=f lambda

V

Is

V is cel v

Vel

And f is frequency and l is lambda

<@&286206848099549185>

Velocity is 50*2.5

Which is 125

Hm

<@&286206848099549185>

[obtuse pebble]

@spark stump Has your question been resolved?

[spark stump]

<@&286206848099549185>

[obtuse pebble]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[spark stump]

Hm

[obtuse pebble]

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[meager sun]

i need help for quadratic equation

[meager sun]

help quadratic equation

[fathom flicker]

Please post a picture of your question

[meager sun]

.close

[obtuse pebble]

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[hidden compass]

Lol

[obtuse pebble]

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[timid silo]

,rotate

[timid silo]

Is number 5 right? I got 0.05mi/s

[obtuse pebble]

@timid silo Has your question been resolved?

[tight patio]

Umm u went wrong somewhere

So ur method is right

Ur working is half Complete

But ur division is incorrect

[obtuse pebble]

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• Show your work, and if possible, explain where you are stuck.
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[reef garnet]

im lost after this bc at 2B^2+4B-9=0 you do quadratic formula and then there are two answers and im not sure which one to pick

also with the one up top, where did this come from???

oh wait im dumb i see my mistake

.close

[obtuse pebble]

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• Show your work, and if possible, explain where you are stuck.
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[vapid gale]

i am having trouble learnign how to graph

[vapid gale]

i was told to put each of these on the graph but i dont know how im supposed to do it

[obtuse pebble]

@vapid gale Has your question been resolved?

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[timid silo]

Someone walk me through this please

,rotate

[obtuse pebble]

@timid silo Has your question been resolved?

[tardy epoch]

Just plug in larger negative and large positive numbers into h

[obtuse pebble]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

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[static beacon]

for question 1

lets say you cut it in half once

your left with 2 peices

do it again 4

do it again 8

[strong fern]

What’s the equation

[static beacon]

you tell me

1,2,4,8,16,32

this doubles everytime no?

what equation could this be

[strong fern]

2n

But what’s the area of each piece

[static beacon]

what happends if she cuts it in half 3 times? whats the total number of peices

[strong fern]

4

[static beacon]

no

[strong fern]

6

[static beacon]

See thats what your equation 2n would suggest

but is that right

[strong fern]

No

[static beacon]

Exactly

we know its 8

how about 2^n

does that work?

[strong fern]

Yea

[static beacon]

we find our equation than

coz if n=0, we get 1, n=1 we get 2, n = 3 we get 8

[strong fern]

So how do I find the area of each piece

[static beacon]

well we know

if we add up the area of ALL the peices, no matter how many times we cut it

it has to be equal to the area of the peice of paper even if it wasnt cut

so tell me whats the area of the paper

when its not cut

[strong fern]

80

[static beacon]

perfect

so if we cut the paper in half, what is the area of each peice

[strong fern]

Still 80?

[static beacon]

it will be 80 all togeahter yes

but whats the area of just one peice

[strong fern]

40

[static beacon]

Yea

Coz we have 2 bits of paper

so if we cut in half 2 times

so we have 4 bits of paper

whats the asrea of each bit

[strong fern]

20

[static beacon]

Exactly

So like our other sequence, instead of doublig each time

it halfs

[strong fern]

So it’s 1/2^n

[static beacon]

Ooo nearly there

but our initial area isnt 1

its 80

so it would be 80/2^n

n = 1, we get 80/2 = 40, n=2 we get 80/4 = 20 n=3 we get 80/8 = 10

[strong fern]

So the equation for area of each piece is 80/2^n

[static beacon]

Yea

that is correct

now question number 2

as you just stated, the area of each peice is 80/2^n

lets call the area of each peice 'A'

we know 80/2^n = A

correct?

[strong fern]

Yea

But isn’t 80 supposed to be divided by 1/2

80(1/2)^n

[static beacon]

hehe thats the same thing

80 * (1/2)^n

80 * 1^n/2^n

80*1^n/2^n

1^n is always 1

80 * 1/2^n

80/2^n

[strong fern]

K

For q2 why is the product and area always the same when u cut it

Idk how to explain it

[static beacon]

well back to what we were saying

we know 80/2^n = A

now if we times both sides by the amount of peices we have after n amount of halves

we basically timsing by 2^n

80/2^n * 2^n = A * 2^n

80 = A * 2^n

they will always equal 80

[strong fern]

So they both just have the same rate of change

[static beacon]

Yea

Well

no

They have the opposite rate of change

one cancles out the other

as one doubles, the other halves

[strong fern]

So I write the product of the number of pieces and the area of each piece is always the same because they have opposite roc

[static beacon]

Yea you could

[strong fern]

K thanks a lot for the help

.close

[obtuse pebble]

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[next lantern]

can someone help me out with part b

[next lantern]

i believe for part a i just add the individual probabilities, but im not sure how to do it with the intersection

wait theyre independent events

would be 0 then?

[brave bramble]

"independent" means that:
P(A n B) = P(A)*P(B)

Note that, for part a), adding them together sometimes doesn't work. Keep in mind the more general formula:
P(A u B) = P(A) + P(B) - P(A n B)

However if {a} and {b} are independent, then adding them is fine.

[obtuse pebble]

@next lantern Has your question been resolved?

[next lantern]

would i be able to use that here since i dont know which a and b share?

[obtuse pebble]

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Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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[violet rapids]

Mr cock

[violet rapids]

,rotate

[thin star]

alright

You didn't have to multiply the 5 with another x-1 / x-1

[violet rapids]

What should I do after that

Then

I mean before that pert

Part

[thin star]

$3 \frac{x-1}{x-1} + \frac{5}{x-1}$ is enough

[warm shale]

TooManyCooks

[thin star]

So from here you can distribute that 3

Once you do, you will end up with two fractions, both with x-1 as denominator

You can combine them together afer that, and you should get the expression you're looking for

[violet rapids]

Wait

so

what

I shouldn’t multiply (x-1) on both sides ?

[thin star]

You only multiply by x-1 / x-1 on the left because will make their denominators equal

[violet rapids]

But whatever I do on one side I need to do the on other tho ?

[thin star]

It's not an equation, it's just an expression

[violet rapids]

Ohhhh

So I can do@thst

Ok

[thin star]

Also, you're multiplying by 1

sort of

[violet rapids]

yes ok ik

ok wait

Don’t tell me the next step yet

[thin star]

Sure

[violet rapids]

U forget I’m stupid sometimes

go slow with the steps

[thin star]

Ok where do you want me to start

[violet rapids]

Do I kepe or standard or in factor

Probablt factor right

[thin star]

I'm sorry what

[violet rapids]

,rotate

[thin star]

Yes that's right

[violet rapids]

Ok

[thin star]

Okay

so now you can distribute that 3

[violet rapids]

,rotate

[violet rapids]

That is not equal to the other equation

I did something wrong

OMG

[thin star]

yes, so when you distribute, you should have multiplied the -1 by 3

[violet rapids]

I forgot to multiply the 3 to -1

yes

[thin star]

😂

[violet rapids]

,rotate

[violet rapids]

good 👍🏼

[thin star]

You can try the same thing for the other problem

[violet rapids]

I’m ok

[thin star]

Okay

[violet rapids]

,rotate

[violet rapids]

That little -2 becomes a positive answer right

Bcuz it’s in the parenthesis

[thin star]

Ah. For exponents, raising to a power is the same as multiplication

[violet rapids]

yes I know that

[thin star]

So it should be -

[violet rapids]

negative

R h sure

[thin star]

I'm positive

https://tenor.com/view/ba-dum-tsss-drum-band-gif-7320811

[violet rapids]

yes u are right

I double checked u

on google

okay

now

ohhhh

U knwo what I should just do for my first step

I could just move the a with the negative little 2 to the denominator right away

[thin star]

Yes! That's actually a great idea

[violet rapids]

too easy for me

I’m too smart

,rotate

[thin star]

You got it

[violet rapids]

aye

u can read side ways too

that’s impressice

[thin star]

https://tenor.com/view/shiba-inu-head-tilt-question-gif-3521016

[violet rapids]

I forgot for this rule

But

Anything to the 0

To the power of 0

Is 1

Right

[thin star]

yes

except 0, that's debatable

i think it's indeterminate if 0

anyway

[violet rapids]

,rotate

[violet rapids]

Can u help me with C

I have to go take a doodoo brb

[thin star]

what do you have so far

[violet rapids]

I erased it

Bcuz I know it’s wrong lol

U can see if

It

Not very erased is it

[thin star]

Oh yeah I just noticed

So you just do the same strategy as before

except now you turn x^0 = 1

[violet rapids]

Wouldn’t that just be 2

[thin star]

Yes

[violet rapids]

OK

[thin star]

not always possible btw, but that's not an issue here

[violet rapids]

I got u

[violet rapids]

I’ll tell my math teacher that

Ok

I got this

[zenith raft]

your nails slay

[violet rapids]

,rotate

[violet rapids]

AW ty

Can I see ur bobs

I’ll show u mine in return

[zenith raft]

you don’t seem old enough soz

[violet rapids]

dang

I got like 1 month

until I’m old enough

[thin star]

You met your match @zenith raft

[violet rapids]

We can have a 3(sum)=product with Mr. cock

[thin star]

[zenith raft]

LOL

[violet rapids]

btw is this the right answer cock

[zenith raft]

lol ok i expected you to be younger for some reason

[thin star]

yes

wait

i thought that had a -1

no that's not right

[violet rapids]

Why

[thin star]

the 2 doesn't have a -1

i misread it

sorry

only x goes to the denominator

[violet rapids]

Ohhhhhhh

I keep thinking they are attached

8/x^2

[thin star]

Yes

[zenith raft]

a wonderful person has just gifted me nitro so now you can see about as much as i can show

[thin star]

[violet rapids]

who gifted u it

did mr cock do it

[thin star]

[zenith raft]

idk if the person wants me to say

[violet rapids]

Omg it was mr@cock

[violet rapids]

okay show

[zenith raft]

i meant my banner is back

[thin star]

Alright let's not get too saucy here. Mods might react

[violet rapids]

R u guys dating

[zenith raft]

this channel is fucking wild LOL

[zenith raft]

no lmao

[violet rapids]

how do I factor this?

[thin star]

Try to look for common factors. Keep factoring them 1 by 1 if it's not obvious

so they both have 3, take out a 3. They both have x, so take out an x. Keep doing that until they no longer have things in common

[violet rapids]

got cha

how do I take out an x if one has an x^2 and the other is just an x

[thin star]

you can factor out the x, but the other x stays on one term, while the other one is left with no x

$2x^2 + x = x(2x+1)$

what the

[warm shale]

TooManyCooks

[violet rapids]

hmm

$2x^2 + x = x(2x+1)$

[warm shale]

Leah

[violet rapids]

hahaha

it likes me more

Ohhh

Like

MOk I get it

I get it but I didn’t really do it

okay

[zenith raft]

get fucked

[thin star]

if you insist

[thin star]

what did you get?

[violet rapids]

Btw is this right

[thin star]

They're both right! Nice one

[violet rapids]

Dude

I’ve been getting so good on the other stuff I started msging u about

I still don’t understand 2 and 8 and my teacher said 2 was wrong

[thin star]

You uh... made some mistakes there

[violet rapids]

wtf where

[zenith raft]

y went poof

[thin star]

Among other things, yeah

By the way, did you change your radical from a cube root to a square root?

[violet rapids]

Which one

I might’ve

[thin star]

Let's start with the first one

You did w correctly

for x how did you get x^3

[violet rapids]

,rotate

[violet rapids]

Is this right

[thin star]

Yes

[violet rapids]

what other one is worntg

wrong

[thin star]

well actually i just noticed you didn't copy the 3

that should be cube root

[violet rapids]

omg

shut up

i hate u

haha

okay

now

which other one

[thin star]

B is almost correct. YOu need to write a 4 on the radical

[violet rapids]

omf

[thin star]

haha

[violet rapids]

same with d

OK

[thin star]

c is good

just noticed b and d are the same

but yes i think that should be it

[violet rapids]

last proble

m

[thin star]

ok

[violet rapids]

Only problem B and C

so I set it up like how it asks me to but

It’s just so weird I think u have to walk me through this

I thought I had the answer right but when I plugged it in

The output wasn’t 27 or 147

[thin star]

Okay. So what do you get when you let x = -2

[violet rapids]

no skip A

I already did that

[thin star]

oh ok

[violet rapids]

that one just plug in

B I need help with

[thin star]

Gotcha

So you're looking for x so that f(x) = 27

[violet rapids]

yes

[thin star]

So you just let $27 = 3x^2$

[warm shale]

TooManyCooks

[violet rapids]

Yep

[thin star]

And you got what

[violet rapids]

the wrong answer

haha

[thin star]

I wanna know what you got. I can't help you if you don't tell me

[violet rapids]

I’m not telling u

[thin star]

[violet rapids]

u can read it in erased form

,rotate

[tardy epoch]

Lol

Wants help

Won't do work to get help

[violet rapids]

I did do the work !!

the wrong work

[zenith raft]

riemann leave leah alone 😡

[tardy epoch]

Orly

[violet rapids]

Riemann helped me pass geometry haha

[thin star]

Fuck off @tardy epoch

[violet rapids]

2 years ago

I remember him

[tardy epoch]

I was trying to help you

But ok

[violet rapids]

Ok so

How do is start with B

For output 27

[thin star]

What did you do first

[violet rapids]

I reverse

I don

Undo the

Wlel

I tried to undo the root

square

little 2

I tried to work backwards basiclaly

to get x alone

[thin star]

Okay, you could do that, but are you comfortable working with sqrt(3)?

[violet rapids]

yes

I did get that

[thin star]

so what do you get after doing that then

[violet rapids]

I divided by 3

[thin star]

Wait what

why 3

[violet rapids]

Look

[thin star]

If I follow your story, you should have $\sqrt{27} = \sqrt{3x^2}$

[warm shale]

TooManyCooks

[thin star]

Ah

When you take the square root, the 3 should also be in there

you didn't include the 3

[violet rapids]

On the right side

?

[thin star]

Yes

[violet rapids]

Got it

Would I leave it like this then

[thin star]

Wait what happened to x

[violet rapids]

Do I still leave the x^2

ok

yes

I do

okay

[thin star]

Why did you divide by 3

it doesn't cancel the sqrt3 on the right

[violet rapids]

yeah but before I didn’t have that on the right side

I’ll erase it

Bcuz I thought it was just going to be 3x

[thin star]

I see

Let me see the new one then

[thin star]

Ok good

by the way, did you guys learn about absolute value yet?

[violet rapids]

yes but I honestly don’t remember that from algebra 1

We did a little bit of that in algebra 2 but not too much

[thin star]

Okay. When you take the square root of x^2 like that, you get a +- sign

You remember the quadratic formula? There was a +- in it. It's the same thing

We can worry about that later, for now I want you to solve for x

[violet rapids]

ok wait

I got a question

[thin star]

Sure

[violet rapids]

There is 2 x intercepts right

Bcuz it’s x^2

A parabola

[thin star]

Not always

[violet rapids]

ok but for this case

yes or no

[thin star]

no

just one

you're talking about f(x)) = 3x^2?

[violet rapids]

no

27=3x^2

[thin star]

oh that

then yes

but I wouldn't call them intercepts

I guess you could

Anyway

we're getting off topic

What do you do now?

[violet rapids]

I got it

But I did it different

[thin star]

That's exactly right

[thin star]

Yes!

[violet rapids]

OK

.close

[obtuse pebble]

Channel closed

Closed by @violet rapids

Use .reopen if this was a mistake.

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[gusty pelican]

are these logically equivalent?

[slate zephyr]

!show

[obtuse pebble]

Show your work, and if possible, explain where you are stuck.

[slate zephyr]

Give your thoughts and we will tell you if they were correct.

[gusty pelican]

okay so

I couldn't really write it down as a proof(?)

bc idk how to do that

and I'm not sure if my professor wants me to explain it with sentences

but here's my thought process so far

I think the first part statement means

All x is P(x) if and only if x is Q(x)

and the second statement means

All x is P(x) if and only if all x is Q(x)

but I can't tell if that's equivalent

all things... are P(x) only if that thing is also Q(x)

all things are P(x) only if all things are Q(x)

so maybe it is equivalent?

[agile oriole]

it is “ for all x”, not “all x”

[gusty pelican]

yeahhhhh so do I sayyyy

for all x....... x is P(x) if and only if x is Q(x)

and then

for all x that is(?) P(x) if and only if for all x....... x is Q(x)

[agile oriole]

Yes

[agile oriole]

you can use parentheses, like

for all x (x is P(x) iff it is Q(x))

[gusty pelican]

∀x(P(x) ↔ Q(x))
for all x (x is P(x) if and only if it is Q(x))

∀xP(x) ↔ ∀xQ(x)
for all x(x is P(x)) if and only if for all x(x is Q(x))

is this right?

for all things,,, thing is P if it is Q

for all things,,, thing is P if for all things,,, thing is Q

they sound? equivalent?

[agile oriole]

iff

[gusty pelican]

is that an abbreviation?

[agile oriole]

@slate zephyr are these correct?

[gusty pelican]

thought it was just a typo oops

[agile oriole]

“iff” means “if and only if”

[gusty pelican]

gotcha

cool

[slate zephyr]

👀

[agile oriole]

I’m not sure, the pinned statement really looks like a tautology to me

[gusty pelican]

I have to see if those two are logically equivalent

[slate zephyr]

yes

[slate zephyr]

yes

but this means it is not the same, right?

[gusty pelican]

is this one more correct than the one you replied to?

[slate zephyr]

in one case you compare the x indivudually

and want them to be equivalent

while in the other case you only want that the statement that all of them are something is quivalent.

[gusty pelican]

sorry which ones are you talking about

I'm comparing them individually in the second one?

[slate zephyr]

yes

Let's consider example

x from {a,b}

P(x) = {true if x=a}

Q(x) = {true if x=b}

Now the first here

[slate zephyr]

^

is undoutably false

because P(a) = true and Q(a)= false

so they are not elementwise equivalent!

However the second statement

[slate zephyr]

^

would be true

because neither P nor Q have all x satisfy something

hence both sides of the quivalence are false and it is equivalent

false is equvialent to false

[gusty pelican]

wait what

the first statement is false and the second statement is true

[slate zephyr]

where

[gusty pelican]

so how are they logically equivalent

[slate zephyr]

$\forall xP(x)$ is false

[gusty pelican]

"Now the first here is undoubtably false"
"However the second statement would be true"

[warm shale]

M8732

[slate zephyr]

$\forall xQ(x)$ is false

[warm shale]

M8732

[gusty pelican]

what do you mean when you say

first statement and second statement

∀x(P(x) ↔ Q(x)) and ∀xP(x) ↔ ∀xQ(x)

∀x(P(x) ↔ Q(x)) is this first statement

∀xP(x) ↔ ∀xQ(x) and this second statement

[slate zephyr]

possibly I was ambigious

yes

this is what I meant

[gusty pelican]

sooo if the first statement is false and the second statement is true

[slate zephyr]

yes

[gusty pelican]

how are they logically equivalent

[slate zephyr]

they aren't

I never said they were, right?

[gusty pelican]

you just said they were

[gusty pelican]

this

?

[slate zephyr]

oh no no

I was talking about the equivalence within the second preposition

[gusty pelican]

oh so you're saying that's what makes the second statement true

[slate zephyr]

yes

[gusty pelican]

so finally

[slate zephyr]

the entire strement is usually not true.

[gusty pelican]

∀x(P(x) ↔ Q(x))
false
∀xP(x) ↔ ∀xQ(x)
true
so NOT logically equivalent

[slate zephyr]

yes

🙂

@agile oriole

Generally a word of advice

Moving quantifiers anywhere usually changes the meaning.

[gusty pelican]

okay I see

thank you

[agile oriole]

Oh, I see now

[slate zephyr]

There is a small number of things that one can do such as reording forall quantifiers (but not moving forall over exists)

I am glad to unconfuse two people at once.

Double efficiency 🙂

[gusty pelican]

could you look at another problem please?

[slate zephyr]

which

[gusty pelican]

I could try translating it into english first

[slate zephyr]

So what is your intuition at this point?

[gusty pelican]

∃x(P(x) → Q(x))
there exists an x(if x is P(x), then x is Q(x))

?

[slate zephyr]

yes

[gusty pelican]

∀xP(x) → ∃xQ(x)
if? for all x(x is P(x)), then there exists an x(x is Q(x))

[slate zephyr]

I think there ∃ different kind of people. I personally always prefered quantifier language over full sentances but I often read that some really like full sentances.

[gusty pelican]

yeah I'm new to this so it doesn't really make sense

[slate zephyr]

Especially in english sentances

I always feel like the bracketing is unclear

[slate zephyr]

^ here

We mean IF

for all x we have P(x)

THEN

there exist x with Q(x)

so like there exists only anything if the entire if is satisfied

it is not an elemntwise statement.

[gusty pelican]

okay so don't translate it like I just did

[slate zephyr]

Oh I think it is fine like you said.

[gusty pelican]

o

[slate zephyr]

I am just trying extra clarify.

[gusty pelican]

ah okay

is the first statement an elementwise statement

[slate zephyr]

yes

or well

it's exist

but we have one x

for everything

because exists first

then bracket

everything inside

[gusty pelican]

okay so

the two statements are not logically equivalent

[slate zephyr]

A steep claim, after I said this

[gusty pelican]

:D

[slate zephyr]

let me actually think about the statement though.

yeah

not true

Now I will let you find P and Q to show they can be different.

[gusty pelican]

uhhhhhh

[slate zephyr]

[gusty pelican]

hold on

[slate zephyr]

No hurry

[gusty pelican]

am I supposed to do it like you did before?

[slate zephyr]

it is probably the eaisiest way

you can also do it with real numbers or something

two elements suffice

[gusty pelican]

can I say phrases like is a duck and is yellow?

or is that crazy

[slate zephyr]

I find it more confusing but you can certainly try.

I mean

P(x)

must be a statement with a paramter.

Perhaps

"x is yellow"

[gusty pelican]

∃x(P(x) → Q(x))
there exists an x(if x is P(x), then x is Q(x))

there exists a thing x(if x is a duck, then x is yellow)

∀xP(x) → ∃xQ(x)
if for all x(x is P(x)), then there exists an x(x is Q(x))

if for all x(x is a duck), then there exists an x(x is yellow)

are the two x's in the second statement separate things?

[slate zephyr]

yes

[gusty pelican]

and x is the same one thing in the first statement

[slate zephyr]

I tried emphasizting tzhat before.

I am not entirely convined you show that it is not equivalent.

really you just started calling P(x) duck and Q(x) yellow

[gusty pelican]

yeahhhh

[slate zephyr]

I think we need a concrete set x and a concrete model where some x are ducks and some are yellow

[gusty pelican]

should I use this and try to explain it?

x from {a,b}
P(x) = {true if x=a}
Q(x) = {true if x=b}

[slate zephyr]

sure

we may need diufferent P and Q, let me see.

I think they will do.

[gusty pelican]

∃x(P(x) → Q(x))
there exists an x(if x is P(x), then x is Q(x))

P(a) = true Q(a) = false
and P(a) if true can't lead to false
so this is false?

[slate zephyr]

no

[gusty pelican]

oof

[slate zephyr]

it says exists

not forall

for x=a it is false

however x=b might... ?

[gusty pelican]

okay I was gonna try that next

[slate zephyr]

oh?

[slate zephyr]

where you nto saying the entire exist is false?

[gusty pelican]

P(b) = false Q(b) = true
and this can be true ?

[slate zephyr]

what do you think?

[gusty pelican]

that part is true

[slate zephyr]

yes

[gusty pelican]

so it is true that there exists

[slate zephyr]

yes

left hand side is true

oh wait

anyway

I will let you finsih

[gusty pelican]

∀xP(x) → ∃xQ(x)
if for all x(x is P(x)), then there exists an x(x is Q(x))

P(a) = true Q(a) = false
true cannot lead to a false thing again...?

P(b) = false Q(b) = true

I'm confused

[slate zephyr]

now not elementwise

first

[gusty pelican]

two separate things

[slate zephyr]

∀xP(x)

is this true?

[gusty pelican]

what is x?

[slate zephyr]

all x

[gusty pelican]

true

[slate zephyr]

a quantifier itnroduces a new variable.

[slate zephyr]

why

[gusty pelican]

can't we just state that for all x x is P(x)

I don't know why O.o

[slate zephyr]

maybe because it is wrong

forall x we have P(x)

would mean that both P(a) and P(b) are true

however P(b) is false!

[gusty pelican]

ahhhhh

that's cool

[slate zephyr]

so what can we say about the entire implication?

[gusty pelican]

it is false

[slate zephyr]

NO

[gusty pelican]

:D

uh oh

[slate zephyr]

false => anything

is true

[gusty pelican]

do we have to check Q

[slate zephyr]

we don't

false implies anything

we only check the second part

[gusty pelican]

but I could check Q to show it

[slate zephyr]

if the first was true

[gusty pelican]

I see I see

[slate zephyr]

show what

[gusty pelican]

that makes a lot of sense

to show that it is true

[slate zephyr]

When you doubt my claim "When it rains, no sun is visible"

and currently there is no rain

[gusty pelican]

but it would be a thorough explanation for a beginner course O.o

[slate zephyr]

then the entire statement - well we cannot verify - but in mathematical logic we say it is true.

[gusty pelican]

is there a name for this way of explaining?

[agile oriole]

or said another way, you can’t find a counterexample from the premises

[gusty pelican]

like setting x = {a, b}

[agile oriole]

it’s called “vacuous truth”

[slate zephyr]

I think it is just the definition of implication.

[gusty pelican]

so if I were to write out an explanation

I wouldn't give a name for the process I did

[slate zephyr]

https://en.wikipedia.org/wiki/Truth_table

[gusty pelican]

and just say say that x = {a, b}

[slate zephyr]

^ here you see what each "logical operator" does given ertain truths and falses.

[gusty pelican]

no I meant what we did here

setting x = {a, b}