help-23

[signal granite]

If P is true for say a cat, then If P is true, Q is true doesnt make sense

[broken yew]

no.

certainly not.

If P is always false

this implication is always true.

==================
If x is a multiple of 15, then x is a multiple of 3. TRUE
If x is a multiple of 3, then x is a multiple of 15. FALSE
If 1 = 0, then 3 = 3. TRUE
If 1 = 0, then 2 = 3. TRUE

Do you understand how each of these work?

Note I use 'then' in place of 'implies'

[signal granite]

Yes but how do those examples follow "If P then Q"

[broken yew]

What do you mean.

You can prove each of these examples by assuming P and showing Q must be true if so.

[signal granite]

I mean like this

P = true but Q is false

how does that adhere to If P then Q

[broken yew]

Your statement is false in general.

You gave a counter example

If B is a cat

we have a contradiction

Hence, B four legged => B is a dog

is not true for all B.

[broken yew]

Ok, technically my first 2 examples are bad. I should say what x is

[signal granite]

so then this holds true if i add an " or P is false"

[broken yew]

Formally, you should use quantifiers in statements

There shouldn't be free variables lying around

You say 'For all animals B, if B is four legged then B is a dog'

(which is false)

[signal granite]

i dont think i get what "If P is true then Q is true" means then

[broken yew]

I don't get what you don't get.

What's the issue here

[signal granite]

me neither tbh its so frustrating ugh? It just wont click

[broken yew]

What exactly is the problem with your example?

'For all animals B, if B is four legged then B is a dog' <--- This is FALSE. We have B = cat as a counterexample

[hollow lance]

Maybe write out the truth table and see if it hits home

[signal granite]

Okey so the way i understand this rn is "If P then Q" is the way to put the statement P => Q into words. But i dont see how If P then Q is equivalent to P => Q

since if P = false Q is true, that does not feel equivalent to "If P is true then Q is true". And if P = true and Q is false it also does not adhere to "If P is true then Q is true"

[broken yew]

What

Ok lets break this up

'If P = false, Q is also true'

??

'If a spider is four legged, then it is a dog'

You are asking about this statement?

Ignoring injured spiders, this is a true statement.

[hollow lance]

I think some semantical examples would help here, like an obligation or contract.

[broken yew]

'If a spider is four legged, then it is a dog'

I'm guessing they are confused why this statement is true

The key is the IF bit.

It is true because there are no spiders with 4 legs.

It is vacuously true you could say

[signal granite]

i rephrased it a bit

[broken yew]

There are no spiders which will contradict this statement. Hence this statement is vacuously true.

[broken yew]

Do you get how this works?

[signal granite]

i get why thats a true statement..

[broken yew]

Ok.

[signal granite]

Im confused on how to put => in words. i dont see how its the same as saying "if p then q"

[broken yew]

But it is.

Give me an explicit example where it does not appear to be the same

[signal granite]

If P = true and Q = false? For example B has four legs => B is a dog (With B being a cat since P = false)

how does that adhere to "If P = true then Q is true"

P is true and Q is not

[broken yew]

Ok lets slow down.

You have

'If an animal has 4 legs, then it is a dog'

As your statement.

You then are concerned about cats in particular

'If a cat has 4 legs, then it is a dog'

[broken yew]

This is false.

[broken yew]

This is false.

What is the issue?

You see, your original statement isn't even true in the first place

[signal granite]

a cat is not a dog and not every animal with four legs is a dog.

[broken yew]

Well yes

So (P => Q) is NOT true

in this case

An animal having 4 legs DOES NOT imply it is a dog

'If an animal has 4 legs, then it is a dog' <-- FALSE

'(P => Q)'

You have to realise this entire thing is a statement

As are P and Q.

You are concerned about 3 statements here.

'P', 'Q', 'P => Q'

[ocean elm]

If an animal has 4 legs => then it is a dog

can be true if it both doesnt have 4 legs and doesnt isnt a dog, no?

truth table for p implies q
p q
t t true
t f false
f t true
f f true

[signal granite]

so @broken yew what you are saying is that P => Q is true, if and only if P is true and Q is true or P is false.

[broken yew]

P => Q is true if:

P is false OR Q is true

This is the logical definition of P => Q

(not P) OR (Q)

'(P and Q) or (not P)' is equivalent to 'Q or (not P)' when you expand it out, I hope you realise

[signal granite]

Okey, But that is not the same as saying if P is true then Q is true

[signal granite]

yea i do

[broken yew]

How is it not the same.

You have 4 cases.

[signal granite]

Because if P is true Q CAN be false?

[broken yew]

Compare the Left and Right in all 4 cases ok?

It CANNOT

[signal granite]

Only in a true implication

but

[broken yew]

If P is true AND Q is false

P => Q is false. (and it's equivalent logical definition)

[signal granite]

So if P then Q is a way to read implications only for true implications

but implications are not necesarily true so how are those two ways of describing the statement equivalent

[broken yew]

The statement

'P => Q' means that the implication is TRUE

'only for true implications' kinda makes no sense

'implications are not necessarily true' ??? I don't get what you men

If I assert 'P => Q', then I am saying this statement is TRUE. ie. this implication is TRUE.

If I assert '(not P) OR Q' I am doing the same thing. I am saying this statement is TRUE.

There is no uncertainty about whether the => is true or not in the statement.

[signal granite]

Quick side question, if a conditional statement is false then is it not an implication?

[broken yew]

What do you mean.

If we have P is false.

Then 'P => Q' is TRUE for ANY 'Q'.

[signal granite]

No thats not what i mean

wait lemme rephrase

[broken yew]

'If the moon is blue, pigs will fly' <-- TRUE

[signal granite]

If pigs are animals is true then the moon is blue. Is this not an implication since the statement is false?

[broken yew]

I'm confused by your phrasing

[signal granite]

If i have a false statement, a conditional statement, is it an implication or not?

[broken yew]

I don't understand how you are structuring that statement

Are you trying to say

'If the statement "pigs are animals" is true, then the statement "pigs will fly" is true'

What you just said like you are questioning

'(A => B) => (C => D)' if this is true or not

[signal granite]

okey so what i understand from what youve been saying here is that if "P=>Q" is true its an implication. If"P => Q" is false then its not an implication but just a conditional statement.

[broken yew]

Or something

"If"P => Q" is false then its not an implication but just a conditional statement."

No???

What is a conditional statement to you?

I'm confused.

[signal granite]

well i thought it was any statement of the form "P => Q" (so the same as an implication)

[broken yew]

'If X, then Y' <-- Are you defining this phrase to always be conditional?

Ok, sure.

I don't really use this terminology

But ok, any statement of the form 'P => Q' is conditional, I agree.

It is possible to have conditional statements that are true. And it is possible to have conditional statements that are false.

Ok.

And then?

If 'P => Q' is true, then we have a true implication

If 'P => Q' is false, then we have a false implication

[signal granite]

Is "If P then Q" a general way of 'reading' implications? or in other ways can any implication be read as "If P then Q"?

[broken yew]

Well yes.

[signal granite]

Okey

..

Why do we read it like that? I cant see why its a valid way of reading "P => Q", or in other words i cant see how "P => Q" is equivalent to "if P then Q"

[broken yew]

Me writing 'A => B' is read as 'If A, then B'. Or 'A implies B'.

No matter if this implication is true or false.

[signal granite]

I can i understand if i think of it as: a meaningfull implication can be read as if P is true then Q is true. (Non meaningfull implications being false implications or "trivial" implications where everything can be proven since the hypothesis P is false)

[broken yew]

The statement reads the same.

Why.

What does 'meaningful' really mean here though

Ok, I can see what you mean

Well if that's how you want to see it.

[signal granite]

P = true and Q = true <=> if P then Q. any other combination of P and Q true-false values either end up false or trivially true since everything can be proven from a false P

[broken yew]

Yes.

[signal granite]

Is it a correct way tho?

Bcs thats the onlyway it makes sense to me

[broken yew]

Well idk?

I don't see how it doesn't make sense otherwise

I agree with you 'If P, then Q' is equivalent to 'If P is true, then Q is true'

That is implied in the spoken language.

[signal granite]

Well you cant read every implication as "If p is true then Q is true" because P=>Q with P = T and Q = F is still an implication but doesnt adhere to "If P is true then Q is true"

[broken yew]

why.

Let me see,

[signal granite]

Well P is true and Q isnt

So if P is true then Q isnt necessarily true

[lean otter]

نعمان

[broken yew]

'If true is true, then false is true'

[signal granite]

Hi

[broken yew]

Is what you've written down.

That clearly is false.

As is the original.

[signal granite]

Yes, but its still an implication no?

And the way to read implications is If P then Q

[broken yew]

Alright, and since when were all implications true

We have true implications and false implications

You gave me a false implication

T => F

Translated to 'If true is true, then false is true'

[signal granite]

omg it clicked

r$guorafsbaQDGRJ.OM GMJ HTKLM

Okey i finally get it

Thank you for being so patient and helpfull!!!!!! ❤️

brb gonna scream ontop of my roof now

[broken yew]

oh really?

good lol

no problems

[signal granite]

Yea it clicked bcs i was about to write If P then Q in this case is false.

thats basically saying its a false implication

so If p then q is a way of describing any implication

anyway.. my brain hurts

[safe radish]

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[woven berry]

I was trying to think of the permutations as 15 digit base 6 number.

[south anchor]

okay so

let's start by saying the red ball is in box 1

which boxes can the blue ball go into?

[woven berry]

4-15?

[south anchor]

yep

and if red is in box 2?

[woven berry]

5-15

[south anchor]

etcetera etcetera right?

so how many ways can you rearrange just the red and blue ball?

just as sum

[woven berry]

There'd be 11 arrangements with the red ball left of the blue ball, right?

so that'd be 22?

maybe?

[south anchor]

so if we have the red ball on the left. if its in box 1, that's 9 places for the blue ball to go right? box 2, that's eight.

etcetera etcetera

[woven berry]

so nPr(9,4)?

then multiplied by 2 since you can switch the roles of the red and blue balls?

[south anchor]

yeah, you need both. so the possible arrangements of just the red and blue ball are 2*(9+8+7...))

whatever that number may be

[woven berry]

78?

[south anchor]

sure

[woven berry]

the sum stops after 4 right?

[south anchor]

nope, stops at 1. the red ball goes up to box 11 but we can only put the blue ball in box 15 then.

[woven berry]

ahh, okay.

so it's 90 then.

[south anchor]

i'm gonna trust your calculator on that one

sounds about right

but keep in mind, these are only the arrangements of red and blue. what about the other three?

[woven berry]

Those can go in any of the non-fixed locations right?

[south anchor]

exactly

so you have 90 combinations for the red and blue, and then each of those 90 has another couple possibilities depending on the three left.

[woven berry]

Once we've fixed the positions of the red and blue balls, there are 13 positions left, can't we just calculate all permutations of a 13 digit base 4 number?

Each of those would represent a location of the other boxes

so that'd be 90 * 4^13?

[south anchor]

i'm not sure where you're getting the base 4 from.

[woven berry]

green, orange, purple, empty

[south anchor]

i don't think so. because the three balls only have one of each. whereas you have 10 "empty" spots.

the empty spots are indistuingishable

[woven berry]

that is true, so we divide by 10!?

nope nvm

that's not an integer

wait, that'd be 13!/10!

right?

13 objects, 10 of which are duplicates, so the number of unique permutations is 13!/10!

[south anchor]

maybe? need to double check my combinatorics equations. that doesn't sound off, though.

i think that's right. and then what's your final answer for the total number of combinations?

[woven berry]

90 * 13!/10! ?

154,440?

[south anchor]

yep! i would double check the logic on all of this, since i'm not always 100% sure. but that seems like a dang good answer to me.

[woven berry]

Awesome, thanks!

.close

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[south anchor]

np

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[fervent mountain]

I am confused on where to start

[fervent mountain]

The question is to find all the roots of 2x^3-3x^2+5x-4=0

[quasi bison]

have you worked with the rational root theorem before?

if so, you can use it here to guess a root and factor your cubic polynomial into the product of a linear and a quadratic.

[fervent mountain]

I have no clue what ur talking about

[quasi bison]

so you have not heard of the rational root theorem then

hm.

[plucky elk]

what class is this for?

[fervent mountain]

algebra 2

[plucky elk]

are you in 10th or 11th grade?

[fervent mountain]

11th

[quasi bison]

well there is no real way around ||guessing x=1 as a solution and using that to reduce your equation down to a quadratic||

[lean otter]

You can factor it by grouping and reduce

[fervent mountain]

how

[worthy hemlock]

You can't factor by grouping

[lean otter]

Yeah you could

[velvet dawn]

Can someone please help with this question ?

[lean otter]

(2x^3-3x^2)+(5x-4)=0

[worthy hemlock]

#❓how-to-get-help

[lean otter]

Take out the x^2

[worthy hemlock]

Still can't factor by grouping

[worthy hemlock]

#❓how-to-get-help

[worthy hemlock]

Continue this process, prove to me that you can factor by grouping

[lean otter]

Sec let me finish my counter strike game

[worthy hemlock]

Did you not read #❓how-to-get-help

Don't ask in an occupied channel

[velvet dawn]

Oh, I’m so sorry. Won’t happen again

My bad

[lean otter]

It only works for some 3rd degree polynomials. This isn't one, my fault

[fervent mountain]

the answer on the review sheet is x=1 and x=1+i √31/4

but it doesn't show the process to get it

[worthy hemlock]

As ann mention, you can see that 1 is solution, so then you can use that to turn that cubic into a quadratic

[plucky elk]

those types of problems are such dick moves

[fervent mountain]

honestly bro can u draw it out and send a pic of it

bc I have no clue what is going on

[worthy hemlock]

Do you know how to do polynomial long division?

[fervent mountain]

yes

[worthy hemlock]

Then that's what you need to do

[fervent mountain]

but it's giving me nothing to divide by

[worthy hemlock]

By inspection, you can see that one of the roots is x = 1

[safe radish]

@fervent mountain Has your question been resolved?

[safe radish]

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[shell stirrup]

Hi

[shell stirrup]

I need help

Hi sneaky

I need help with very basic variational calculus

gime a sec

So the activity here is,
i wana prove that the stationary/minimum path between two points in a cartesian coordinate system is a line

So my math book defined y(x) as the minimal path

and using this y(x) , they add an arbitrary function of x called g(x) thats multiplied by a real parameter ε

And what they want to do is to vary ε
until it gives the curve equation of minimum path between A and B

So this is what i mean ^ This derivative expression here

Where S is the functional representing the curve length between A and B

[shell stirrup]

but since they already defined y(x) as the minimal path

like they want S[y + ε g ] to take its minimum at ε=0
coz S[y] gives the minimum length of curve i.e. corresponding to minimum path

but like
my 1st question is : tf is the point of bringing up ε in the first place if we are just gona zero it? it feels very redundant

and my 2nd question is :
If i were to use an arbitrary function t(x) that creates an arbitrary path between point A and B rather than using y(x) which represents minimal path immediately, can we also work towards showing t(x) = minimal path

this da whole page

<@&286206848099549185>

[safe radish]

@shell stirrup Has your question been resolved?

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[amber hemlock]

I need help with q7

Here’s the answer^

[final loom]

Yk PY and PX are given

tan PYX can be calculated I believe

PYQ is just double PYX... PXQ can be done in the same way

shaded area is calculable once you get PYQ and PYX

[amber hemlock]

Ure the same dude that helped my friend HAHAHA

[amber hemlock]

Wait but I can’t assume it’s 90 degrees tho

[final loom]

If not, then XY is also given in your question

Use cosine law

$\cos PYX = \frac{7^2+5^2-4^2}{2\cdot5\cdot7}$

[flat frigate]

Ansh

[amber hemlock]

I tried it

And multiply by 2

But it’s not the ans 😦

[final loom]

,calc 2arccos(0.82857)

[flat frigate]

The following error occured while calculating:
Error: Undefined function arccos

[final loom]

,calc arccos(1)

[flat frigate]

The following error occured while calculating:
Error: Undefined function arccos

[final loom]

,w 2arccos(0.82857)

[flat frigate]

Results provided by WolframAlpha

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

Requested by Ansh#3502

[final loom]

,w 2arccos(0.7142857)

[flat frigate]

Results provided by WolframAlpha

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

Requested by Ansh#3502

[final loom]

The answers are correct except a) and b) are inverted

a) is 1.19 and b) is 1.55 and not the otherway

[amber hemlock]

OH

Ok tysm

Ok I think I can do it now

Thanks again

.close

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[umbral meteor]

Is this also correct? I saw the answer is 2/3pi in the answer book

a is representing cosinus btw and b is representing sinus

[crisp lodge]

you could check that using a calculator

[umbral meteor]

Oh okay

[crisp lodge]

but it would even be more useful if you knew the period of tan function

[umbral meteor]

It’s just using the unit circle

[umbral meteor]

Ok so my calculator told me -60° so convert that to pi and it’s -1/3pi so my answer is also correct right?

[crisp lodge]

if the calculator says so

then yea

[umbral meteor]

Okay thx for the the tip to check on calculator btw

[crisp lodge]

well sure, but using the unit circle, what period would you get for tan function?

[umbral meteor]

• k times 2pi?

[crisp lodge]

hmm

2pi ?

[umbral meteor]

Sorry pi

[crisp lodge]

yep exactly

and -pi/3 +pi = ?

[umbral meteor]

So -1/3pi + pi would give me that

Ah thx

[crisp lodge]

yea

And actually it would be more accurate if you wrote your solution like that

unless you're given a defined interval

[umbral meteor]

Oh okay i will do it like that from now on then

[umbral meteor]

Yes then i’ll just do it without period

[crisp lodge]

yea exactly

if the question asked for a solution in the interval of 0 < x < pi, then your answer would not meet the requirements

so only 2/3 pi would be the answer

[umbral meteor]

Ah so then

Yeah

I get it now completely

Thx a lot!

[crisp lodge]

nps :)

[umbral meteor]

tysm for your help i really appreciate it! Have a nice day 😄

.close

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[zealous hawk]

How many triples of (a,b,c) there are, if we know that abc=10^9?
Ps: 1,2,3 and 3,2,1 are counted as two separated triples

[zealous hawk]

help

please

[hushed anchor]

are there any other restrictions?

like does a and b and c need to be distinct

[zealous hawk]

yes

a,b,c ∈ N

[hushed anchor]

i cant think of how to subtract the overcounting but if a, b and c are not distinct it should be 2025

[zealous hawk]

answer is 3025

i just brute force it threw coding

but i need a prove 😐

[hushed anchor]

oh lol

i was thinkinf of using sticks and stones

(2^r)(5^x) * (2^s)(5^y) * (2^t)(5^z)

then r+s+t=9
x+y+z=9

[zealous hawk]

um

wha?

first, what is sticks and stones?

[hushed anchor]

oh um

its hard to explain

[zealous hawk]

is it combinatorics part?

[hushed anchor]

yeah

[zealous hawk]

oh

its just called diff at my country

balls and partitions 😄

[hushed anchor]

its like find how many possible sollution are there for a question like
a+b+c=9

[hushed anchor]

yeah lol there are many names

[zealous hawk]

ok so

but there's multiplication

as well i think its obv that we have 100 dividers of 10^9

so now we have to somehow inmplemenet it into our task

[hushed anchor]

i mean a, b and c are all gonna be either 1 or multiples of combination of 2 and 5

[zealous hawk]

yes

10^9=(2^9)*(5^9)

[hushed anchor]

so i thought of assigning
a=(2^r)(5^x)
b=(2^s)(5^y)
c=(2^t)(5^z)

[zealous hawk]

yep

r+s+t=9?

[hushed anchor]

yeah exactly

[zealous hawk]

x+y+z=9?

[hushed anchor]

yeah so each has 10 choose 2 sollution

so 10 choose 2 =45

and 45 square is 2025

but idk that doesnt seems to be close to the sollution

[zealous hawk]

i would rather belive you than this 😐

[hushed anchor]

nvm i found my mistake yes we

yes its 3025

11 choose 2 not 10 choose 2

but still there is repeating which we havent accounted for

[zealous hawk]

bruh

i forgor formule for this thing

[hushed anchor]

like the first one here
has 1 repeated twice

[zealous hawk]

ye

so empty spaces allowed

right?

[hushed anchor]

wdym empty spaces

[zealous hawk]

like here

empty space in middle

and at the end

nvm

[hushed anchor]

well its not 4 bars but yeah empty spaces jus mean ^0

[zealous hawk]

ye

so C out of 11 by 2?

or what

im lost a little bit

[hushed anchor]

yeah that splits up th epowers

adn you gotta square because there are two prime divisors

[zealous hawk]

ye i got it

thanks!

[safe radish]

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[lean otter]

my brain is not working im doing some geomerty make up work rn

[lean otter]

i have no idea if i got the answers to 1-9 right and i am stuck on the proof at 10

<@&286206848099549185>

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[signal granite]

Kinda stuck on basic mathematical logic concepts.

In an implication of the form P => Q the truth value of Q is independent of P. Then what relationship does an implication describe and why is it necessary/usefull to define it.
(To me it seems like P => Q only informs us on Q if you know the truth value of P and of the implication as a whole. If P = True and P => Q = True then Q must be true. But then how can you deduce the implication AND Q to be true given only a truth value for P (direct proof method))

[broken yew]

'the truth value of Q is independent of P'

How. Of course not.

If P is true, then Q MUST be true.

That is clear dependence.

[signal granite]

For a true implication right?

[broken yew]

???

Well yeah?

[signal granite]

In my book the text says that an implication is not causal thats why i thought that

[broken yew]

I mean if you're giving me a statement and telling me it might or might not be true

it's kinda meaningless

Hi, 0 = 1

Might or might not be true.

But think about it

???

idk how to put this.

'P => Q' asserts the implication is true when it is stated.

[signal granite]

could you maybe explain what an implication is from 0. Im just gonna reset my memory of what it should be

[broken yew]

???

You read 'P => Q' exactly the same as 'if P is true, then Q is true'

Take this definition OR

$[P\implies Q]:=[(\lnot P)\lor Q]$

[flat frigate]

Shuri2060

[broken yew]

Take either definition and convince yourself they are equivalent.

[signal granite]

im dropping out of this damn class

Anyway, that definition makes sense and i see why theyre equivalent. But it only makes sense to me if the implication is causal

[broken yew]

bruh

[cyan flame]

Discreet math?

[broken yew]

you're overthinking things by a mile

or something

I really don't get what you don't get

[cyan flame]

Takin it this semester it's hell but satisfying AF

[broken yew]

what do you mean casual ???

what's a non-casual implication

[signal granite]

causal not casual

wait maybe thats not a word in english, i kinda just took the dutch word and went with it lol

lemme google

It is! "relating to or acting as a cause."

[cyan flame]

P=>Q means that if P is True then Q must be true for eg
If it's sunny I will definitely go out
This means
Being sunny => I go out

[lean otter]

I think he meant by 'independent' that each propisition is separated from the othet like
(a+b)^2>_0 implies |x| >_0 for a,b,x in R

[signal granite]

Yeah, or like 1+1 = 2 ==> the moon is made of cheese

[broken yew]

can u do latex

[lean otter]

nope

[broken yew]

_0

what

[lean otter]

greater or equal

[broken yew]

ok ok

[broken yew]

no

thats false.

[lean otter]

Yeah

But is that an implication

[broken yew]

You can write whatever the hell you like. But that might make the implication (the entire statement) false.

[signal granite]

Okey so then how can i prove Q to be true if i only know P is true

[lean otter]

consider 1+1=2 ==> 8-1=7
is that an implication or it isnt cuz they are separated

[signal granite]

its an implication

[broken yew]

Can we have a concrete example

or some clarification

[signal granite]

If P => Q, but i dont know the truth value of ANY statement except P with P = True. How do i prove Q = true

[broken yew]

So you have

P AND (P => Q)

Correct?

And utilising this, you wish to prove Q is tue

We start with the logical definition of implication

[signal granite]

no without knowing the truth value of P => Q

[broken yew]

???

[lean otter]

u can prove it even if you dont know the truth value of P

[broken yew]

Then what the hell

[lean otter]

just use a table of truth

[broken yew]

If you don't know the truth value of P => Q

Then Q is clearly independent to P in general

So of course you cannot prove Q

[signal granite]

but what im describing is exactly the method of "direct proof"

[broken yew]

It is not.

[signal granite]

you take P to be true and then prove Q somehow without knowing if the implication is true. thats the whole point you prove Q and the implication using P only

[broken yew]

The method of direct proof

well depends what you're referring to

Can we, again, have a concrete example

However simple, idc.

[signal granite]

uhhmm lemme think of one

[broken yew]

Suppose $1 + 1 = 2$.\

This implies that $1 + 1 + 1 = 2 + 1$.\

Simplifying the right, we have $1 + 1 + 1 = 3$. QED

[flat frigate]

Shuri2060

[broken yew]

Will this do?

We have 'proven' the statement

$$(1+1 = 2)\implies(1+1+1=3)$$

[flat frigate]

Shuri2060

[broken yew]

Note I used some axioms there in the process (or this would be impossible, obviously).

@signal granite What's the issue here

[lean otter]

They arent seperated

[broken yew]

huh?

[signal granite]

i get what youre saying but im still confused, that didnt clarify it. Idk y (im about to smash my head against the wall omg)

[broken yew]

Your issue seems to be that:

In order to prove 'P => Q' is true, we start our proofs by assuming P is true and attempt to show Q is true.

===
Yes or no?

[lean otter]

$((a+b)^2>0)\implies(|x|>0)$

[flat frigate]

ihab

[lean otter]

a b and x r in R*

well both propositions r true but they are seperated

[signal granite]

I guess? i understand why we assume P to be true (for it to be a meaningfull implication, meaningfull being a non false hypothesis since you can prove anything from that)

[signal granite]

yeah i feel like you get it

[broken yew]

Is this example what you were thinking of?

I can proceed to show this in the normal way as well

[signal granite]

like what use is the implication, it doesnt give us any information idk

[broken yew]

That's fine though

Because there are axioms

and stuff outside we use in order to show it

Let $a, b, x\in\bR$.\

Suppose $(a+b)^2\geq0$.\

Since the absolute value function is always non-negative, we have $|x| \geq 0$.\

We assumed $(a+b)^2\geq0$ and showed $|x| \geq 0$. Therefore, for all $a, b, x\in\bR$, we have $((a+b)^2\geq0)\implies(|x| \geq 0)$. QED

[flat frigate]

Shuri2060

[broken yew]

This proof works fine.

[lean otter]

yea

[broken yew]

You see, 'P => Q' is 'P is true implies Q is true'.

It is not 'we need P to be true for Q to be true'.

The difference between necessity and sufficiency

Maybe that's the confusion?

[broken yew]

The latter would actually be 'Q => P'. 'We need P to be true for Q to be true' is another way to phrase 'Q implies P'.

[lean otter]

Maybe you should prove that they are equivelent first then prove the implication which is easy as we have equivelence (double implication)

[signal granite]

hmmm okey that cleared up alot in my brain. Ill go ponder this, i hope i'll understand in a few.

[broken yew]

So yeah

===========
'P => Q'

'P being true is sufficient for Q to be true'

'Q being true is necessary for P to be true'

All 3 are equivalent statements.

And of course, 'P <=> Q':

'P being true is necessary and sufficient for Q to be true.'

[signal granite]

Like idk y i dont get this. during the semester i was fine, i completely got it. heck i even did good in logic class but now im stuck on something so basic

i hate my brain sometimes

[broken yew]

Well yh

'if and only if'

Did confuse me a lot when I first thought about it

'only if'

🤔

[signal granite]

if and only if is an equivalence

<=>

[broken yew]

Yes, but translating to english the 'if' and 'only if' ----- I was confused with which was which. Might be the same thing you're confusing.

[signal granite]

ig i was saying (P=>Q) <=> "If P then Q"

and its just confusing how any implication can be read that way since P=>Q is true if Q = true, regardless of P.
To me it only makes sense to read an implication as "If P then Q" for a true implication with P = True (a "meaningfull" implication)

[broken yew]

yes.

P being true or false is independent of Q

If we know 'P => Q'

'For all complex $x$, $x$ is real implies $x^2$ is non-negative'

[flat frigate]

Shuri2060

[broken yew]

When we add quantifiers, things make more sense imo

Because then we need the P false cases

$$(\forall x\in\bC)((x\in\bR)\implies(x^2\geq0))$$

[flat frigate]

Shuri2060

[broken yew]

When we check this statement, we check it for real x (in the proof)

The entire statement is true, despite x not always being real

[signal granite]

an implication is true if and only if (P then Q) is true

Yeah okey

I think i get it, just gotta ponder it a bit

But i think my whole perception//thought structure of reading logics is off

since "P^Q" is also read as P and Q

In other words P^Q is true only if P = true and Q = true

but P = False and Q = True still is P^Q and is still read as P and Q

Do you get what im trying to say?

The way we read connectives is based on their truth value. P^Q is read as P and Q bcs its only true when P is true and Q is true

For P or Q to be true either P or Q has to be true so its read as P or Q

An implication is only true (meaningfully) if P then Q so its read as if P then Q

so ig i get it but the meaningfully part is still throwing me off a bit

[broken yew]

(I'm reading and trying to get your thought process)

ack. I don't know if I follow 😅

Is your first language not english btw?

if not, then it will be harder to express even more hahaha

[signal granite]

No its dutch haha

[broken yew]

yh lol ok

[signal granite]

but even in dutch i wouldnt be able to explain bcs this whole implication bs is killing my brain

Anyway thanks for the help and patience!

Im gonna try and clarify what i mean a bit more so i can put how i understand implications in words better

.close

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[lean otter]

For 1c , it would just be 1<t<3?

Bc slope is negative right

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@lean otter Has your question been resolved?

[lean otter]

<@&286206848099549185>

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@lean otter Has your question been resolved?

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@lean otter Has your question been resolved?

[light shoal]

In this context, does "speed" have the usual physics meaning of "absolute value of velocity", or does it just mean "velocity"?

[lean otter]

usual physics?

im not sure

[light shoal]

The answer will be different depending on the meaning of speed. Since part (a) mentions velocity and (c) mentions speed, I'm guessing that they are not the same thing in this context, but you should check with your instructor to be sure.

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[proper pier]

if part (ii) is asking for an equation like y(x,t) = Acos(kx-wt+phi), are parts (iii) and (iv) just asking me to substitute in a value for x?

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@proper pier Has your question been resolved?

[proper pier]

shoot I did not realize that was actually someone's username ok

[proper pier]

alright whatever this isn't even graded on accuracy

.close

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[tame prawn]

were you the one who pinged me? lmao

[proper pier]

yes I apologize 😔

[tame prawn]

its fine, i don’t really get annoyed by pings

i have my notifications turned off either way

[proper pier]

fair enough

have a good day

[tame prawn]

you too 👋

[worthy hemlock]

FYI you opened a channel and you can close it using .close

[tame prawn]

oh

ty

.close

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[naive quarry]

Can someone help me with this, I don’t know what to put x and y as

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Please don't occupy multiple help channels.

[naive quarry]

<@&286206848099549185>

[naive quarry]

Ok

[pulsar condor]

@naive quarry .

[naive quarry]

Oh

,close

.close

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[edgy ridge]

I tried to find angle between 21 and 30 using law of cosines

then I found the height using sin(angle)*21

then I did height*width/2

how did this not work

hmm seems like i made an arithmetic error

.close

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[fast widget]

did I mess up anywhere?

please @ me

[broken yew]

desmos?

[zinc crown]

,w area enclosed by 2/x where x > 0, 8x where x > 0 and 1/2x where x > 0

[elfin yew]

(unbounded)

[fast widget]

wtf

[zinc crown]

(unbounded)

[fast widget]

how do I use desmos to check?

[zinc crown]

he was talking about the graph

but u got the graph right

[fast widget]

Yeah I need to find the Area

and that's the part that I am unconfident about

[zinc crown]

i dont know wtf wolfram did

oooh it found the negative branch

[flat frigate]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by Yottachad#6969

[zinc crown]

also if that is webassign

just submit it and then tell us if you got the wrong answer

and if its a test...you should delete ur question cuz if a mod sees you'll get banned

[fast widget]

It's webassign

[zinc crown]

so just submit ur answer?

[fast widget]

yeah it's wrong

we have submission attempts

/5

[zinc crown]

alright

ur integral looks right so lets have wolfram do it

,w int from 0 to 1/2 of (8x-x/2)dx + integral from 1/2 to 2 of (2/x - x/2)dx

[flat frigate]

Results provided by WolframAlpha

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Requested by Yottachad#6969

[zinc crown]

yeah guess ur integral was wrong

[fast widget]

yeah I'm kind of bad at integration. Can you tell me where I messed up?

[zinc crown]

here for one thing

1/2x -> 1/4x^2

not 1/4x

[fast widget]

I'm about 0.1 off of wolfram's calculations @zinc crown

[zinc crown]

almsot

ur forgetting to distribute the negative

the - (-1/4*1/4) is a +1/16

then u have the right answer

[fast widget]

I need sleep

[fast widget]

thanks

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[mint totem]

<@&286206848099549185>

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@mint totem Has your question been resolved?

[mint totem]

<@&286206848099549185>

[plucky elk]

What have you tried

[mint totem]

wait

i will share

This is what I did

[leaden folio]

for question a) you should see how many km does it take to go to the cafe in the way back. how would that help you to answer the question?

[mint totem]

it would take 14950

to get back to cafe

but they travelled extra 10

so they missed it by 10

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[young swift]

Using Taylor Series Method approximate y(2) given that y'=x+2y, y(1)=1 correct to three decimal places

[split ether]

What's confusing you here?

[young swift]

i dont know how to approach this ques

my topic is weak, so if i could get a step solving for this, it wud be good

[split ether]

Notice, from the information given, we can solve for linear approximation of y at 1

[young swift]

yeah, i see, then

[split ether]

So the approximation would be y(1) + y'(1)x, right?

[young swift]

yess right

[split ether]

y(1) is just one as given

Now what'd y'(1) be?

[young swift]

y'(1) would be umm

[split ether]

y'(x) = x + 2y(x)
=>
y'(1) = 1 + 2y(1) = 3

[young swift]

im sorry my brain is fogged rn

oh yeah right

[split ether]

Yeah, plugging the values in gets us that the linear approximation of y near 1 is just 1 + 3x

[young swift]

ohh yeah, now i get it

[split ether]

Now we want the approximation at 2, which is just substituting x = 2 into the approximation

[young swift]

so 2+2y(2)

thatd be 6?

[split ether]

The approximation is 1 + 3x, so at x = 2 it is 1 + 3*2

[young swift]

ohh shit, my bad

righttt

so at 3, it wud be 1+9

10

[split ether]

Yes

[young swift]

and is that the answer to our question?

[split ether]

But the question asked for approximation near 2, so that's irrelevant

Yeah it's just 7

[young swift]

thanks my friend! helps a lot

i had a test tomorrow, so was practicing final topics

thanks!

[split ether]

Np, feel free to .close

[young swift]

.close

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[inland jackal]

2x + 3y = 1
5x - 4y = -32
What does y equal pls help

[inland jackal]

Im close to ending it all

[quasi bison]

@inland jackal are you seriously considering committing suicide right now?

[inland jackal]

No but if I dont get this maybe

NOT LIKE I WOULD

[quasi bison]

if you're suicidal then you need to call a suicide hotline.

@undone palm please do not give out answers.

[inland jackal]

Im not gonna harm myself I promise

[undone palm]

sorry

[quasi bison]

ok, so now that that's out of the way

just to make sure

[inland jackal]

[quasi bison]

you typed the second equation in your system as 5x - 4x = -32. is this intended, or did you mean 5x - 4y = -32?

[inland jackal]

I intended 5x - 4y = -32

[quasi bison]

okay

that's a typo fixed then

so, have you worked with systems of linear equations before, or is this your first time solving one?

[inland jackal]

First time

[quasi bison]

okay... so you have not yet been introduced to methods such as substitution or elimination?

[inland jackal]

No

[quasi bison]

hrm

well, ok

i could explain some general principles that go into solving a system of equations no matter its form

[limber lynx]

The way I like to approach these questions is to get two different values for one variable and then equate them

[quasi bison]

to hopefully make the method of elimination (and others) less opaque than they would otherwise be

[inland jackal]

Ah okk

[quasi bison]

the most important principle of solving systems of equations is that the less variables appear in one of the equations, the better.

the ideal case is when you have an equation in one variable only, which you can use to solve for said variable and plug it in everywhere to get rid of it entirely

this is usually achieved through combining the equations of your system in some way. exactly what "combining" means depends on the method used.

does this principle make sense to you?

[limber lynx]

Since you want y, let's isolate y.
3y=1-2x
y=(1-2x)/3
Also, 5x-4y=-32
4y-5x=32
4y=32+5x
y=(5x+32)/4
You know what to do from here

[quasi bison]

...

heisenberg i would rather you hadn't done that

[inland jackal]

Yes

[ancient escarp]

in the middle of your in depth attempt to assist, i will hereby answers instead!

[quasi bison]

okay so let's continue

i am going to illustrate the method of elimination on a different example

[inland jackal]

Ok😁

[quasi bison]

here is a different system:

s + 7t = 30
-s + 2t = 6

[pure pendant]

either isolate x or y, or play with sums of two equations until you isolate one x or y on one side and a value on the other 😄

[quasi bison]

.-.

[inland jackal]

OH

[quasi bison]

why do i keep getting interrupted

[pure pendant]

well, sorry haha

[ancient escarp]

as the prophecy foretold

[pure pendant]

i find it might be a little easier if i explain how you generally do this in one saying

[quasi bison]

i was getting there

[limber lynx]

yes

[pure pendant]

sorry for interupting you, i'll be watching out to avoid that in the future 😅

[quasi bison]

speed of explanation is not a metric to optimize like this

[quasi bison]

i was going to continue by drawing attention to the fact that something very interesting happens when we add the two equations together in this

namely:

s + 7t + (-s) + 2t = 36
which simplifies to 9t = 36, an equation in one variable only

this is important

[inland jackal]

Yes

[quasi bison]

like, this ties back into the principle i stated earlier that you want to get equations with less variables, ideally with only one

for the method of elimination, the way you do this is by adding or subtracting two equations in your system in such a way that the terms with one of the variables cancel out

[inland jackal]

Im starting to get it

[quasi bison]

in my example, the s terms cancelled out, so i have eliminated s

it usually does not matter much which variable to designate for elimination; however, if it happens that you're being asked not for the solution of the whole system but only for the value of one of the variables, it makes sense to eliminate the other.

so like, in your case you are asked for y, so it makes sense to eliminate x. you could eliminate y instead, solve for x and then use that to solve for y, and this would not be illegal, but it would be roundabout.

note that elimination might not be as neat in other cases - sometimes the coefficients on the variable to be eliminated are different in the two equations

like, coming back to your system

2x + 3y = 1
5x - 4y = -32

we might want to eliminate x, but the coefficients on x are 2 in the first equation and 5 in the second - so it won't work if we just try to add the two together

to fix this we can multiply both sides of one (or both) of the equations by some number

so we could multiply both sides of the first equation by 5 to get

10x + 15y = 5

and both sides of the second [equation] by 2 to get

10x - 8y = -64

and voilà, now we have the same x term in both equations, so we can subtract them

do you see how to continue from here?

[inland jackal]

Im kinda slow let me just examine that for a second

[quasi bison]

no worries, take your time

i tried to go in as much detail as i could, but if there is something i said that doesn't make sense then please let me know

[inland jackal]

OH I SEE

Where did the -64 come from

[undone palm]

she multiplied 2 to both sides of the equation

[inland jackal]

OH YEAH

[quasi bison]

@undone palm i'm a she.

[undone palm]

so sorry

[inland jackal]

Ok after that step whats next

[quasi bison]

well now that we've made it so x can be eliminated

we eliminate x

by subtracting the two equations

10x + 15y - 10x + 8y = 5 - (-64)

[inland jackal]

@mortal slate watch

Ok so we eliminate x which is 0 correct?

[quasi bison]

we aren't saying x itself is zero

we don't know the value of x, and we don't care,

eliminating it means making it so it no longer appears in an equation

[inland jackal]

Oh ok

[quasi bison]

the whole point of this multiplying by 5 and 2 in our starting equations was so that we can do this

[inland jackal]

Yes

[quasi bison]

after doing some cleanup on this, what do you get?

[inland jackal]

7y

[quasi bison]

you start with an equation, you do some cleanup, and you end up with something that does not have an equals sign in it?

besides, the left-hand side doesn't simplify to 7y either

[inland jackal]

[limber lynx]

@quasi bison can i step in?

[inland jackal]

I understand everything except the subtraction part

[quasi bison]

well i'm made vaguely uncomfortable by the presence of an emoji titled "tyler the homophobe" so idk heisenberg maybe you can

[limber lynx]

haha it's prob ironic but ok

[inland jackal]

OH NO IM SORRY I CAN DELETE IF YOU WANT

[limber lynx]

well which step exactly?

[inland jackal]

IM NOT HOMOPHOBIC THO

[inland jackal]

Subtracting both equations

[limber lynx]

i see

well what do you think 10x-10x is?

[inland jackal]

0

[limber lynx]

does that mean x=0?

[inland jackal]

Yeah

[limber lynx]

think again

why would it mean x=0?

say x=2
10 * 2 - 10 * 2 =0

20-20=0, after all

so whatever the value of x is, 10x-10x=0

[quasi bison]

10x - 10x is 0 regardless of the value of x

[limber lynx]

yeah

[inland jackal]

Ooh ok

[quasi bison]

the whole point is we don't need to know the value of x to cancel out 10x with -10x

[limber lynx]

analyze the left hand side

[limber lynx]

yep

10x-10x+15y+8y is another way of writng the left hand side right?

i did nothing but move around the -10x term

[inland jackal]

So now I subtract 10x-10x and 15y with 8y?

[limber lynx]

youre half right

you evaluate 10x-10x

but you dont do 15y-8y

why would you?

it's 15y+8y right?

[inland jackal]

Yes

So it stays the same

[mortal slate]

HELP

[limber lynx]

what would 15y+8y be?

[inland jackal]

23y

[limber lynx]

correct

23y+0 is equal to?

[mortal slate]

mind if i ask a question?

[limber lynx]

sure

[inland jackal]

23y

[mortal slate]

how do you add and subtract integers.

for example -8 + 5 ik ik its an easy question i just forgot.

[limber lynx]

correct now that is the lhs after a bit of cleaning up

[limber lynx]

one minute ill help

[plucky elk]

#❓how-to-get-help

[mortal slate]

WHOEVER REPLIED W THE CAT EMOJI U ARE ON UR DOWNFA-

[limber lynx]

now the left hand side (which you've simplified to 23y) is equal to 5-(-64)

right?

are you following

[inland jackal]

SROP

Yes Im following

[limber lynx]

yes so what is 5-(-64)?

[quasi bison]

@mortal slate read #❓how-to-get-help and get your own channel

[inland jackal]

69?

[limber lynx]

correct

now we have 23y=69

what's y?

[inland jackal]

Y=3?

[limber lynx]

and there's your answer

all we did is

move stuff around

by moving stuff around in a way that's convenient to us

[inland jackal]

OMG

[limber lynx]

we get the answer

easy, right?

[inland jackal]

YES

TYSM

[limber lynx]

no problem

[inland jackal]

IM NO LONGER CRYING

[limber lynx]

that's good

math is no reason to cry

ik it can be tough

[inland jackal]

Yes

[safe radish]

@inland jackal Has your question been resolved?

Channel closed

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[rigid venture]

do we have to factor the -1 out?

[rigid venture]

why do we have to multiply it by -1?

[split ether]

We don't have to, it just looks nicer and simpler to factor

[rigid venture]

yeah but then the roots are gonna be different

(x+1)(x-3)

[thin bridge]

the roots will be the same if you do it properly

what's

(x+1)(x-3)
supposed to be and what steps did you take to get it

[rigid venture]

ohh a is a negative

alright

factoring only works if a is 1 right

[rigid venture]

1 + (-3) = -2 and 1 times -3 is equal to -3

[thin bridge]

factoring only works if a is 1 right
no, its just a little more complicated if it isn't

and you may need to do more work

[rigid venture]

you have to multiply -1 to 3 first right

[thin bridge]

there are multiple routes that you could take

[rigid venture]

wait, that's not the only one?

[thin bridge]

well you could factor out -1, (or if you have an equation multiply both sides by -1, like they did)

[rigid venture]

alright

thanks

.close

[safe radish]

Channel closed

Closed by @rigid venture

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

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!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[novel magnet]

would the instantenous velocity be -0.16?

[sonic sail]

as b gets closer to 12 on the interval [12 b] what number does the slope of secant line approach

[novel magnet]

hmm

well its -0.0016

[sonic sail]

-0.0016 the last value is clearly greater than -0.16

but notice how the numbers never go above zero

[novel magnet]

right

[sonic sail]

yet they grow bigger

[novel magnet]

would it just be -0.0016 then?

[sonic sail]

the goal is to make b approach 12

why stop at 12.0001 when you can choose b=12.000000001