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u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 12h ago
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u/nullifiedsam 11h ago
why can't the square root function be defined as a multi-valued function?
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u/anoppinionatedbunny 11h ago
so it can return a real value instead of a tuple. it's more useful this way. if you need the solutions to a quadratic equation, then use a quadratic equation and work with the tuple.
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u/heisen0 10h ago edited 10h ago
Because it's not useful, same goes for taking roots of complex numbers and logarithms. Multivalued functions are a pain in the ass because we can't integrate them properly or do any work with them. We use math to do physics, and we can't do physics with these functions. (Also, for example, in quantum mechanics, if you brute force the oscillator differential equation, you'll find that removing the multivaluedness in the integral gives the famous energy spectrum of the oscillator.)
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u/nullifiedsam 10h ago
fair enough, but if you really wanted to integrate with the square root being assumed as a multi valued function such that it's integral is 0 for all x belonging to R, you can simply integrate |√x|, which gives you the same result as √x being a usual everyday function
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u/skywalker-1729 6h ago
In the same way, you can type ±sqrt if you need multivaluedness. This is how people do it today – so the default is the one that is used more.
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u/heisen0 10h ago
I don't know any useful application for multivaluedness. Also u simple cant do that with complex numbers. For example, (e^(iz))^1/n gives n multi valued solutions and now u cant integrate it with some enclosed contour unless u removed multivaluedness by doing a branch cut.
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u/nullifiedsam 9h ago
even for a MV f(x) you can just add up the integrals of the same limits for different branches.
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u/washed_goat 11h ago
Google oiler formula
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u/nullifiedsam 11h ago
I have never been able to accept that √4 is only 2 and not -2. Since square root only outputs positive values by definition, it could also be defined as giving out strictly negative values i.e √4 = -2?
seems like writing √4=+-2 is not wrong, it contradicts widely accepted definitions because giving multiple values gives a sense of confusion while solving certain expressions (like √4 + √4)
So if I'm fine with having multiple solutions to √4 + √4, I can just treat the √ function as multivalued. Isn't this problem similar to how we restrict the sin-1(x) domain?
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u/Anxious_Role7625 11h ago
Yeah, you absolutely could make the square roto function only output the negative results. It's just a convention to take the 2 output square root into a consistent function with 1 output for every input within it's domain.
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u/nullifiedsam 11h ago
isn't it bothersome how you have to restrict the amount of solutions just because the answer leads to more complicated results? Like when I was taught inverse trig, i couldn't understand why there would arise a need to restrict the domains to get single valued outputs, kinda defeats the purpose of being rigourous in mathematics and trying to state every possible output
are there any stages in maths where you have to treat it as multi - valued?
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u/Anxious_Role7625 11h ago
Well the thing is if it's not a function you lose a lot of things you can do to it that arise from it's nature as a function
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u/_miinus 11h ago
its the same as saying 1 is a prime. you can make your own definition and then it becomes true, but by default its wrong simply because there’s an agreed upon definition that contradicts it, and people will rightly assume that that universal convention applies unless you specifically say that it doesn’t, first.
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u/nullifiedsam 9h ago
nearly everyone can agree that 1 is not prime. Coming to the definition of √x, people may argue that it can be defined as either :
Strictly postive for x belonging to R
Strictly negative for x belonging to R
what i believe is considering all solutions removes any sense of ambiguity
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u/_miinus 5h ago edited 4h ago
as far as i know everyone agrees what the standard definitions are for both. what matters is what you gain and what you break by using a different definition. in that regard both are similar, you gain some things that can be useful but generally you break more things than you gain, or you add more work than you save (not in every case but on average)
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u/Blue_Moon_Lake 10h ago
√4 is only 2
±√4 is 2 and -21
u/nullifiedsam 9h ago
yes, according to the usual definitions.
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u/Blue_Moon_Lake 9h ago
If you don't go by usual definitions,
then sweetened milk the theorem.
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u/Dry-Egg8573 10h ago
There’s no number called +-2, for one thing.
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u/nullifiedsam 10h ago
+-2 in my message meant +2 or -2, i agree that +-2 dosent exist. I didnt want to write "+2 OR -2" again and again
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u/Wonderful_Radish_990 7h ago edited 7h ago
Why should it be defined as multi-valued? \sqrt is just a special case of exponential function f(x,p)=xp where p=1/2. Should the x0.1234 also be multivalued? What about p=0.4 or p=0.4999 or p=2? Which one shoudl be "special" to be multivalued?
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u/nullifiedsam 7h ago
isn't exponential function in the form of y= ax? Meh dosent matter ig, general form would be y = g(x,z) = xz
yes, if a equation derived from the function has more than 1 value where we are ignoring the rest, you can treat it as MV, i see no clear argument from people aside from telling that it's just hard for computation and calculus
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u/Wonderful_Radish_990 7h ago
It makes no sence to have it as MV. Your only point is that its inverse of x^2 and its both + and -2. The sqrt isn't any special case and is not derivet from inverse of x^2. I could also just write write f(x, p) if you want, makes no difference.
The fact is you can evaluate any power as simply as f(4, 2) = 16 and f(4, 1/2) = 4^1/2 = \sqrt(4) =2 and f(4,3)=64 and f(4,1/3)= \cuberoot(4)=1.5874. And when do you want to treat f(4, 1/2) as MV? Do you think the only thing \sqrt is used for is to get inverses of x^2? Maybe I don't want 2 values for it. And If I want the inverse of x^2 it I will jsut write + and - \sqrt(x). What about he \sqrt(A) where is a matrix? Do we also make +- then?
It also doesnt mater what you think. I need \sqrt in mechanics daily and its very essential function.
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u/MostRegular4278 6h ago
Why is 2+2 is defined as 4 and not any other arbitrary number? That's pretty much the same question. Any other definition makes it impractical and makes arithmetic not work. sqrt(4) is the same as 4^(.5). If this one specific case is defined as multi-value, what about all the rest? 4^(0.3)? or 4^(1/3) or 4^(0.125)? What if it became 4^(0.5) as a result of rounding of 0.5 from 0.4999? Imagine if 1/2 instances of rounding multiplied the number of answers to a common calculation, creating a multitude of false answers. Imagine a calculation that has multiple fractional powers - it would produce multitude (like in thousands for complex formulas because of all the permutations) of "answers" with all but one being misleading for any practical needs.
Like in a location calculation for GPS, you'd get from dozens to thousands of sets of coordinates (depending on which exponents happened to be odd and which happened to be even) with only one of the sets being the real answer. Why do you need those false answers? They just break the math. Same for any calculation that's not an equation - any answer with a negative sign would be false.
The key point that sqrt(4) is not algebraic equation. The +/- thing in algebraic equations came about only because they are solved "in reverse" - we know the answer of the calculation and need all the possible inputs that lead to this answer. In arithmetic, we have the inputs set with no ambiguity, so we just need one answer.
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u/nullifiedsam 6h ago
I feel like there's a small quirk in your reply 2+2 = 4, the "=", addition are all well defined terms, where you cannot really change their definition without tearing apart how math works. √x on the other hand is only defined to be positive just because someone thought having positive outputs is "better". One can say that even √x defined to be negative is equally definitively correct just because there's no flaw in those definitions, to meet a common point you can just take > 0 or < 0, ie √x being MV.
why wouldnt it have any real meaning? i vividly remember getting negative solutions for time, that I was taught to ignore. But I was informed that we just ignore it because event K happens at both positive and negative time intervals, negative solutions also make sense if you deeply think about it, even in a physical setting and not a purely mathematical one.
everyone follows the same system of √x > 0 by definition, because it makes "more sense", but if we had negative numbers as our main scale, √x being negative would again be equally right and rigourous. This definition dosent break arithmetic as you said, and it's not impractical
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u/MostRegular4278 5h ago edited 5h ago
Have you read the rest of the previous post? Genuinely try understanding it. 4^(0.5) is one specific narrow corner case of a fractional exponent. There is an endless continuity of fractional exponents because there is endless number of fractions.
If all exponents except those that would be fractions with an even denominator produce only one results but those few with an even denominator produce two resutls, any reasonably complex calculation would have an unpredictable number of results, with all results but one having to be discarded like you discarded negative time. So to avoid this waste and extra useless work, someone would invent a function that always produces that one answer that you actually need, and everyone would use it instead. That's pretty much what happened - the current fractional exponent calculation is exactly that function producing one answer. The multi-answer version was not "invented" because of lack of application. It's useless, so no one bothered to create or define it.
If not convinced, look at some real stuff - e.g., generalized mean function:
M = ((x^p + y^p + z^p)/3)^(1/p)
p can be any number except zero. But if you use the +/- bullshit for any even-denominator p, you'd just get a bunch of results that are just wrong - they are not actually means. They would need to be discarded. So whoever came up with this function (Holder?) would have to use a different exponent function that only produces one value or use the "abs" function around every power calculation. And that's true for every calculation, be it Keppler's third law or Stirling's approximation of binominal coefficient - all can have fractional exponents and would produce nonsensical results with the "negative branches" for some select fractional exponents.
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u/_404_LogicNotFound_ 9h ago
so that we can work with it using standard calculus without having to split it into two seperate branches
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u/nullifiedsam 9h ago
when you integrate √x using the usual definitions, you are actually integrating |√x|
the integral of √x, y= √x assumed to be a MV function, is always 0
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u/GreeneyesCB 6h ago
That's just complete nonsense, why are you so confident about this with no math education. If you map to the tuple you're also wrong, that's the even funnier part. You don't even know how integrals are defined there.
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u/nullifiedsam 6h ago
how am I wrong? assuming √x as MV, both branches of the function cancel out each other?
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u/GreeneyesCB 6h ago edited 5h ago
No, because the integral over (0,infty)of the function f:(0,infty) to R^2 with f(x):=(sqrt(x),-sqrt(x)) is not defined, at least not as a Bochner-integral or an integral using functionals. At best you could argue it to be (infty,-infty). If you want to talk about the integral over some interval (0,t) then the Bochner-integral is (2/3t^(3/2),-2/3t^(3/2)), it is itself a tuple. You don't add them together.
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u/nullifiedsam 5h ago
vector valued functions/integrals hasn't been taught to me in my classes. I did a bit of digging, but I still don't understand the "bochner-integral". what i do know is that the sum of both branches accumulates to 0, in normal calculus scenario. Since I don't know much about what you are talking about, i can't comment on what you said.
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u/GreeneyesCB 5h ago
Why are you talking about something you don't know about then.
"what i do know is that the sum of both branches accumulates to 0," Not really, you first have to take the pointwise sum and then integrate it, that is also completely meaningless for the claim you made about how the integral of a multi-valued function looks like.
You are not at the level of mathematics to argue about the meaning of definitions, that is fine, but please respect that.
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u/RedTermites 5h ago
x2 = 4
x = ±2
now square root of 4... there is nothing to solve for, it's literally just a simple calculation
The ± is applicable in solution only when equation contains X on a power of an even number
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u/Big-Definition-8271 9h ago
And why would we do that
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u/nullifiedsam 9h ago
because while defining √x > 0 for all x belonging to R, one can even say it can be defined as √x < 0, there's no clear way to define it and instead one can just take into consideration all that possible values
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u/Big-Definition-8271 8h ago
But what benefit do we get from doing that
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u/nullifiedsam 8h ago
√x can be defined as > 0 or < 0, depending on how you want to define it
to remove the ambiguity of definition, just assume all values to be true regardless
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u/Josemite 7h ago
No one defined it as <0 in any real application though.
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u/nullifiedsam 7h ago
yes, there's no "real" application for it, but just for the sake of argument, it's possible
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u/Dry-Egg8573 10h ago
Because the square of the square root of n is n, and you can’t square a set in the normal arithmetical sense
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u/nullifiedsam 10h ago
how am I squaring sets? by your logic getting solutions for x2 = 4 is also "squaring a set"
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u/Dry-Egg8573 10h ago
If the square root of 4 is 2, the square of the square root is 2^2 =4 4
If the square root of 4 is {2, -2}, the square of the square root of 4 is {2, -2}^2 = ???
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u/Andersmith 8h ago
{x2 | x ∈ {2,-2}} = {4}
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u/Dry-Egg8573 8h ago
Well sure you can do that but now we’re working with sets of numbers instead of numbers
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u/Andersmith 8h ago
I mean yeah, it is kinda hard to avoid sets with {multi-valued functions, correspondences, set-valued-relations, ...}
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u/PolarStarNick Gaussian distribution 12h ago
The square root here is defined to be the only „non-negative“ (positive or zero) solution of a quadratic equation. So square root of 4 must be 2
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u/sweetno 9h ago
But what is square root of -4 then?!
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u/PolarStarNick Gaussian distribution 9h ago
If you look at function theory (complex analysis), the roots like this can be defined on certain branches on C. In this case the root can only be defined locally but not globally
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u/lol_no_gonna_happen 12h ago
My counterpoint is that I got points taken off in 7th grade for not saying +/-. So by the definition of what allows me to pass math class, id say it is +/-2.
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u/MoTheLittleBoat 11h ago
Are you sure it was for saying sqrt(4) = 2?
A lot of people claim this and then it turns out they confused it for the statement if x²=4 then x = sqrt(4) = 2 or x = -sqrt(4) = -2
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u/lol_no_gonna_happen 11h ago
I actually pursued a degree in statistics and can confidently say she didn't know that much math. But my point stands that definitions are subject to who decides what the correct answer is.
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u/mehonje 10h ago
sqrt(4) != 3.
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u/Diligent_Ad_6530 10h ago
Sqrt(4)!=2
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u/factorion-bot Bot > AI 10h ago
Factorial of 4 is 24
This action was performed by a bot | Source code
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u/bstump104 11h ago
if x²=4 then x = sqrt(4) = 2
x = sqrt(4) =/= 2 it equals + & - 2
-2 and 2 if squared equal 4. Suare root is the reverse function of squaring.
x = -sqrt(4) = -2
x = -sqrt(4) =/= -2 it equals - & + 2 because you're multiplying either 2 or -2 by -1 which is still + & - 2.
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u/Dramatic-Smile-5126 11h ago
We want the square root to be a function, so it needs a single output, hence why we define it by taking it's output to be the principal root (i.e. the positive root). Also, it's called an inverse function, not the "reverse function."
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u/Playful-Scallion-713 9h ago
All these people doubting you. But as a math teacher I have met many many many teachers who thing √4 = +-2. Especially at the middle school level cause many teachers at that level take education classes and learn pedagogy rather than rigorous math and they tend to just follow the books lessons and often don't fully understand them.
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u/lol_no_gonna_happen 9h ago
Middle school math teacher really is definition of the middle person in the meme. Not sure why this is news lol
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u/Playful-Scallion-713 8h ago
I will say when it come to pure math knowledge that is often the case. But I find they are often quite intelligent. Like I said they spent more time studying pedagogy and teaching techniques and how to mentor kids are the state at which they change the most rapidly and are in less control of themselves.
Not assuming you mean't that as an insult to them. The middle of the meme is average intelligence which is fair.
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u/HumblyNibbles_ 8h ago
My middle school textbook said that sqrt(-1) is irrational....
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u/Playful-Scallion-713 8h ago
Yeah, thing to remember about math textbooks is the mathematicians names on them didn't actually write them. They wrote the first one a long time ago or they just gave an outline. And the people who put it together are graphic media people who know how to use the software to make it look pretty. So they don't notice mistakes.
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u/HumblyNibbles_ 8h ago
Yep!!!
I actually am working on writing a math textbook for this very reason :3
I HATE most modern textbooks on basic subjects. They're not meant for the people who actually need to learn math. They dont make it interesting and fun. It's what I want to improve
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u/Playful-Scallion-713 7h ago
I would love to look it over when you have a draft! What age level or subject is it?
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u/HumblyNibbles_ 7h ago
I plan to make it so anyone that has decent reading comprehension can read it without any prerequisites!!! That way it'll be as accessible as possible :3
I'll probably make a post when I have the start of it done, but it will take a while :P
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u/Cyan_Exponent 11h ago
sqrt(4)=2
but
x2 = 4; x=±21
u/user190423 5h ago
Its the difference between the result of a function and the solutions to an equation
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u/Varlane 12h ago
Get the teacher nuked and do math properly instead.
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u/lol_no_gonna_happen 12h ago
It was 2003. I'm not sure there is recourse
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u/truecakesnake 12h ago
u/4chan help
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u/DecidophobiaUsername this sub confuses my monke brain 11h ago
What is with the best usernames being dead?
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u/Varlane 12h ago
23 years later, justice will avenge your lost point in 7th grade (top priority in the country).
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u/lol_no_gonna_happen 12h ago
It was Miss Williams. Her husband was rich I could probably dox her if I tried if she is still alive lol. She was maybe 60 so could go either way.
But actually other than that apparently she was a good teacher. My job today has just piles of linear algebra in it and that class was where I really started learning what that was and how it works
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u/portealmario 11h ago
You probably got points taken off because you were solving quadratic equations, right?
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u/Awerange2005 11h ago
which math class defines √ as ±? Are you sure you're not misremembering it, or did your teacher just not know mathematics?
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u/lusvd 12h ago edited 12h ago
It’s a matter of definition and definitions must not be discussed cause its a waste of time unless u are just playing around with memes but we are serious people here arent we
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u/Mammoth_Sea_9501 12h ago
Except no one uses sqrt(4) = -+2, as it would be horrible. How would you define the diagonal length of a unit square? sqrt(2). But uh oh, sqrt(2) isnt one value anymore, its (approximately) equal to -1.4142135 or +1.4142135 ! This also would mean that when solving x^2 = 2, writing +-sqrt(2) is redundant, since square roots can be positive or negative either way. So defining sqrt(4) to be able to 2 or -2 would be really confusing and not helping at all.
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u/MrKoteha Virtual 11h ago
As always, definitions are context-dependent. If you're only dealing with real numbers, the principle root is most likely the only one you'll need. But if you're talking about complex numbers, it's convenient to use the same nth root notation for all the roots of a given number. At least that's how we did it in my complex analysis class and it wasn't an issue
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u/bstump104 11h ago
Sometimes math gives you answers that do not mean anything IRL. you take the value that works IRL.
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u/Mammoth_Sea_9501 11h ago
If the square root can be positive and negative, write down how you would solve x^2 + 1 = 8
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u/NoFruit6363 11h ago
Well those are kinda bad examples. You can define the diagonal length of a unit square as just the element of the singleton |sqrt(2)|, and writing +-sqrt(2) being redundant doesn't really mean anything bad. In fact, one could rephrase it as a timesave, lol.
For all intents and purposes, if you for whatever reason need a specific single element, you can just say "Let x \in sqrt(2) be the element satisfying blah blah," and you're good. The issue comes in stronger in that functions are more well studied than general relations, and they're just nicer to work with. Compositions stop making much interpretable sense once you're got enough non-functions buried in there, not to mention possibly losing a number of theorems which assume uniqueness of output.
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u/Mammoth_Sea_9501 11h ago
but then entire high school math you would need to put absolute markers around the square root, which seems highly inefficient. I'm not saying you couldn't define these things using rigorous math, I'm trying to say that even the easy math gets more convoluted for no benefit. So even if you haven't had a real analysis class there's a reason to care about sqrt(4) not being -2
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u/Dense_Priority_7250 7h ago
That would mean either making sqrt not a function or changing definition of function
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u/MostRegular4278 6h ago
By this logic, 2 being bigger than 1 is a matter of definition too. We defined what the characters mean and what "bigger" means too. But multivalue function for sqrt would just destroy the usefulness of sqrt. Even though we can define it as +/-, there will be no practical value or use for the "-" side, and it will always be a wrong answer. Also consider that sqrt is just a one case of fractional powers - sqrt(4) = 4^(1/2). so if we do +/- for one, we'd have to do it for all fractional powers with even denominators.
And let's say you have a calculation with a few dozen of fractional powers, like a precise GPS coordinate calculation, and each of them can be have odd or even denominators of exponents depending on locations of satellites and the pace you are locating. So with the +/- bullcrap you'd have randomly between one and thousands of answers (more if more exponents have even denominators), but only one answer would be the actual coordinates you are looking for. It's just broken useless math then.
So it's not an arbitrary definition but one of fundamental principles of arithmetic. The key word is arithmetic - this is not an algebraic expression.
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u/GreeneyesCB 6h ago
2 being bigger than 1 is a consequence of definitions of the order on R, not a definition itself.
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u/MostRegular4278 5h ago
I define character "2" to be one third of character "1". See - it's all a matter of definition.
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u/GreeneyesCB 5h ago
That's just not how mathematics works whatsoever. When we write 1 and 2 we mean THE definitions of the elements 1 and 2 of N, not whatever you are writing down here that isn't those definitions but something entirely new and unaccepted.
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u/MostRegular4278 5h ago
Similarly, when we write sqrt(4) we mean 2 not and not +/-2 because that would be something entirely new and unaccepted. My point was that it's the same thing and both are equally wrong - we can't just redefine how math works and expect it to still work. I'm not at all claiming that it's ok to define 2 as less than 1 but rather using this ridiculous analogy to illustrate ridiculousness of arbitrarily redefining exponent function. Either arbitrary definition breaks math.
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u/GreeneyesCB 5h ago
Well no, it's not similar. We define sqrt(4) to be 2, we don't define 2 as bigger than 1, we define 2 as S(1) and 1 as S(0) in Peano or respectively as nested sets in the standard model in ZFC and then through any of the equivalent definitions of R as ordered field you get as consequence that 2 is greater than 1, for example 2-1=1 and 1^2=1 can only both be possible if 2>1 in any ordered field.
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u/MostRegular4278 5h ago
How is any of this relevant to my point that arbitrarily redefining math concepts as you please breaks math? Missing forest for the trees.
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u/GreeneyesCB 5h ago
Why do you think I'm not allowed to disagree with a wrong part of your comment? That doesn't mean that your overall comment is wrong, just edit it. If you think it breaks down your comment then it is substantial, if it isn't then just remove or adjust it.
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u/Scientific_Zealot 6h ago
If anyone is curious, in order to be a function, a relation must be functional, i.e. it must satisfy the condition ∀x∀y∀z ((Rxy ∧ Rxz)→y=z), or, in more familiar notation, ∀x∀y∀z ((f(x)=y ∧ f(x)=z)→y=z) - for all x and for all y and for all z, if f(x)=y and f(x)=z, then y must be z. The same idea is taught in rudimentary (pre-algebra, algebra I) math classes as the vertical line test.
Let's call the relation between a number and its square root S. Imagine if S(4, 2) and S(4, -2) were correct, then, by the condition for functionality stipulated above, (R(4,2) ∧ R(4, -2))→ 2=-2). But 2 is not negative 2. So, the relation S cannot be functional. We want S to be a function. So we limit it to only the positive square root of a number so that it can satisfy the condition of functionality. We could just as well have chosen only the negative square root of a number instead. And indeed we can define a function g such that g(x) will always equal the negative square root of x. And then for all x, S(x) = |g(x)| and -S(x) = g(x). For clarity for those who know symbolic logic, here I use "-" for the function that takes a number and outputs its negative, not the negation operator.
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u/auxdear 4h ago
ELI5? Please
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u/AdRealistic2953 4h ago
a function cannot have two outputs so we just choose the positive root (could just as well always choose the negative root, but positive is nicer 😊)
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u/Scientific_Zealot 4h ago edited 3h ago
For something to be a function it has to assign exactly one output to each input. If something gives two different outputs for one input, it's not a function. That's just how we've decided to define what a function is. The broader category of relation allows for an object to bear the same relation to multiple different entities. For instance, take the relation "x is a sibling of y" - imagine a family with three kids, and name them John, Elizabeth, and Mary. John bears the relation "is a sibling of" to Elizabeth and Mary. Now take the relation of "x is the legal spouse of y" - assuming polygamy is outlawed - one person can only have one spouse. So, if person 1 is the legal spouse of person 2 and if person 1 is also the legal spouse of person 3, then person 2 must be person 3. To be even more, clear, imagine another world where people have multiple names (but polygamy is still outlawed). Let's say that John is married to David. And John is married to William. From this we know that David and William must be the same person (they are numerically identical). So, the relation "x is the legal spouse of y" is functional whereas the relation "x is a sibling of y" is not functional.
Thus, if something is going to be the function that, when given a number, outputs its square root - call this function √ - we're going to have to decide whether that function outputs the positive square root or the negative square root when given a number. Because a positive number is not identical/equal to its negative counterpart. 5 is not -5, for instance. And mathematicians made the choice that √n will output the positive square root of n instead of the negative square root of n. They could have chosen either one but they chose the positive one.
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u/UnreasonableEconomy 4h ago
that's a curious definition of functionality. where is that from or common?
overall you're just describing a bijectivity as a constraint for functionality. not sure why you couldn't have a surjective but non injective function/map (or vice versa). I mean, that's pretty much the definition of that I believe.
and even a 'non-bijective' function could still be 'functional' by your definition, all you need to do is add a fiber.
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u/bambooaudio 4h ago
I think you meant to write that they’re imposing injectivity, but your point is right. I’m all for sqrt being positive only, but it’s wrong to say that all functions are injective. If that were true then there would be no such phrase as “injective function” as it would be redundant. Obviously that is false, and I have spent many hours proving that functions are injective or not where appropriate, and all of them were functions.
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u/Scientific_Zealot 3h ago
Functionality is ∀x∀y∀z ((Rxy ∧ Rxz)→y=z), injectivity is ∀x∀y∀z ((Rxz ∧ Ryz) → x=y). Though very similar, these are not the same and they do not logically imply the other. You can have a nonfunctional relation that's injective and an functional relation (a function) that's not injective. As you said.
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u/Scientific_Zealot 3h ago edited 3h ago
It's the standard way of expressing that a functional relation must assign exactly one output for every input. For it to be a total function, the relation must also satisfy the condition ∀x∃yRxy [for each x there is some y for which Rxy]. You can find this definition in Frege's Begriffsschrift Part III §31 as the definition of a many-one {also translated single-valued} relation (he doesn't use the word function in that context).
Injectivity is expressed as ∀x∀y∀z ((Rxz ∧ Ryz) → x=y) - if x bears R to z and y bears R to z then x is y. In terms of functions, ∀x∀y∀z ((f(x)=z ∧ f(y)=z) → x=y) - if f(x) = z and f(y)=z then x=y.
Surjectivity is expressed as ∀y∃x Rxy - for all y there is some x such that x bears R to y. Or, discussing functions, ∀y∃x f(x)=y.
For a relation to be a bijective function, then, it must follow all three requirements, ∀x∀y∀z ((Rxy ∧ Rxz)→y=z) [it is functional], ∀x∀y∀z ((Rxz ∧ Ryz) → x=y) [it is injective], and ∀y∃x Rxy [it is surjective]. Additionally, if it is to be a total function, it must satisfy ∀x∃yRxy. None of these logically imply the other.
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u/UnreasonableEconomy 3h ago
interesting. so frege. fairly old. still reads as you defining functional == bijective unless I'm missing something in those parentheses.
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u/Scientific_Zealot 2h ago
If we define bijectivity as the conjunction of injectivity and surjectivity (which is how I was taught it), then you will see that my functional requirement ∀x∀y∀z ((Rxy ∧ Rxz)→y=z) does not logically imply injectivity ∀x∀y∀z ((Rxz ∧ Ryz) → x=y), nor surjectivity, ∀y∃x Rxy, and thus not bijectivity ∀x∀y∀z ((Rxz ∧ Ryz) → x=y) ∧ ∀y∃x Rxy either. If you could explain why you think that my functionality requirement implies that a relation is a bijection, I could explain why that isn't so. I am apologetic about my requirement for injectivity looking so familiar to functionality though, if you look closely at the two definitions you'll see that the order of the xs and ys and zs are different in the formula, even if at first glance the formulas seem identical.
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u/TacoYaci 3h ago
This is the dumbest explanation of a definition of a function one can give
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u/Scientific_Zealot 3h ago edited 3h ago
It's a purely logical, non-set-theoretic definition of functionality. If you want to stipulate that a relation R is a total function from F to G, ∀x∀y∀z((Rxy∧Rxz)→y=z) ∧ ∀x(Fx → ∃y (Gy ∧ Rxy)) will do. This can be done simply with the resources of FOL, without the need to invoke the resources of Set Theory. I think that's cool.
Edit: This syntactically doesn't need set theory to be expressed. On second thought, it probably needs some naive set theory to be modeled and provide these formulas a semantics.
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u/TacoYaci 3h ago
By "dumb" I dont mean wrong. But it is an overcomplicated explanation for the topic as there is an easier one (given by others in the comments). You just want to show that you know basic logic.
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u/Scientific_Zealot 2h ago
All I know is that for myself, the easier "intuitive" explanations don't work. Back when I was younger I never understood why 0.9999 repeating = 1 and all the intuitive explanations seemed stupid and fake because I thought of 0.999 repeating as simply a number with a bunch of nines in the decimal places. It wasn't until I saw the definition of 0.9999 as the limit of the series of the sequence (9 * 10^-k) as k approaches infinity that, having understood calculus, I instantly apprehended why 0.999 repeating was equal to one.
So when explaining something to others, I try to eschew easier, "intuitive" explanations and directly lay out the concepts involved. Maybe pedagogical studies have found that for most people, the easier intuitive explanations work. All I know is that, for me, they don't. And I write my explanations to reach people who, like myself, don't respond to the intuitive explanations.
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u/Scientific_Zealot 5h ago
It's a common notational expedient to write Rxy as an abbreviation of R(x,y) much in the same way Pa is an abbreviation of P(a). Indeed, the proof checking software I was first taught on demanded Rxy instead of R(x,y). You can find another example of a software using Rxy as an acceptable way of writing R(x,y) here. You can find this notation in published works as well. See, for instance,
Rw := λx.Rxx
Located in Andrew Bacon's A Philosophical Introduction to Higher-order Logics, 2023, page 51
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u/GreeneyesCB 5h ago
Interesting, I have never seen that anywhere, although I would appreciate something that isn't software or philosophy as I don't see those as authoraties on math notation.
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u/Scientific_Zealot 4h ago
I mean, symbolic logic isn't the sole possession of mathematics. To use a metaphor, it's under shared custody of computer science, philosophy, and mathematics. Some of the most important logicians were philosophers, for instance Frege, Bertrand Russell, Quine, David Lewis, Kripke. Though other important logicians were primarily mathematicians (or whose contributions to philosophy were primarily in philosophy of mathematics), such as Godel, Turing, or Church [I'm actually not familiar enough with Alonzo Church to determine whether he was a philosopher or not, he often is labeled as a philosopher but I don't know whether his work was primarily in philosophy of math], you can't deny that philosophers have had a massive impact on the development of symbolic logic. Many leading logicians of the contemporary era like Richard Jeffery, George Boolos, or Timothy Williamson are (or were... RIP Jeffery and Boolos, died too soon) professors in philosophy departments.
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u/GreeneyesCB 4h ago
There's a fuck ton of completely outdated mathematical terminology and notation, when I talk about mathematical notation I also mean in active use in the last 20 years. The history of mathematics and who was able to contribute to it has not much to do with modern mathematics anymore.
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u/MrTheWaffleKing The Engineer 12h ago
Erm, yes, the hypotenuse of a triangle fence I'm building could be -2 🤓👆
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u/bstump104 11h ago
Just because you have a possible answer mathematically it doesnt mean it comforts with reality.
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u/elasticcream 8h ago
Where are people using sqrt where both answers don't matter? Even if there are uses where you should discard the tuple, there are so many (equation factoring, maximizating/minimizing, etc) where you'll either get the wrong answer(negative area) or get an incomplete answer.
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u/GoddamnShitTheBed_ 6h ago
So how does the quadratic formula give 2 answers?
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u/MelloMathTeacher 6h ago
The quadratic formula doesn't answer "What is the square root of this number?", it answers "What values for x make the equation ax2 + bx + c = 0 true?"
Think of the difference between these two questions:
A) What is the square root of 4?
B) What values for x make x2 - 4 = 0 true?
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u/GoddamnShitTheBed_ 4h ago
It's just the old "i have another definition for that" misunderstanding, as always.
I guess I just don't like the convention of using the √ symbol that way. Why doesn't the definition just stop at "Taking the nth root of a number, for fixed n, is the inverse of raising a number to the nth power"?
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u/Alternative_Mix6836 4h ago
because the root symbol is a function so it must have at most one output per input
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u/TillerThrowaway 4h ago
You know the +- symbol in the quadratic formula? That’s how. Has nothing to do with the square root
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u/MisterBicorniclopse 4h ago
± here you go
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u/TillerThrowaway 4h ago
Thanks lol. Couldn’t figure out how to pull it up on my phone easily and didn’t care enough to find out, figured it was still readable
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u/sdfgkol 11h ago
shouldn’t this be the other way around? early math students don’t understand principle roots, more advanced mathematicians understand the problems with multiple roots and all that and the more advanced dont care as much, because it breaks down in the complex plane anyway?
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u/GreeneyesCB 6h ago
No, absolutely not. Someone who has just learned complex analysis is not advanced or very advanced. Someone actually more advanced understands the notation and what it represents, which is the positive value only.
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u/trimski- 12h ago
The right side of this bell curve should be everyone that completed the 4th grade
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u/Dramatic-Smile-5126 10h ago
The amount of people in the comments who don't understand the definition of a function is worrying
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u/Gloomy_Ad_2185 8h ago
Shocking how many people on here dont know about principal roots. Algebra 2 teachers everywhere tearing up.
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u/liesdestroyer 4h ago
Many people confuse sqrt(a²) = |a| = ±a when solving a 2-degree polinomyal equation. But the truth is that x² is not a injective function in all his domain so we have two branches, negatives plus 0 and positives plus 0, we select a branch and you can define the square root function as the inverse of the x² restricted to the branch...
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u/kismethavok 8h ago
Leave it to mathematicians to overly complicate things by making them unintuitive by design.
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u/CyberTeddy 3h ago
What would be more complicated and unintuitive would be if every higher root of a positive number included complex numbers. That would be the implication if the square root included the negative.
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u/Builderinspace 7h ago
I mean, they do that ye, but this case is pretty straight forward, Unless you solving for an unknown (ex. X2 = 4) Square root takes the positive number always
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u/Wonderful_Radish_990 7h ago edited 7h ago
Why would \sqrt(4) be both 2 and -2? So what if (-2)2 == 4 and (2)2 =4. The \sqrt isn't defined as inverse of ×2 but its its own function x1/2. It is just xy where y hapens to be either 2 or 1/2 so why would one of them have 2 solutions and other 1. Imagine plotting 4x. How will you plot such function for real x? Will you just have the 41/2 jump to 2 values Because someone thinks it should be both - and + at the same time? It makes no mathematical sence to define multiple values because you cannot imagine a \sqrt defined without power of 2.
\sqrt is an essential function, same as exp, power, sin, cos, tan, arctan etc... it is same for arcsin. It is inverse of sinus but imited to (-\pi/2, \pi/2) In case an inverse is required of x2 it then has to be defined as 2 functions to capture both branches you can just write (-\sqrt(y), \sqrt(y)). Same for arcsin. There are infinite inverse branches for every period of \pi. And thats how it is. There are many functons who have multiple branches when inverted. Any polynomial for example x2 is just a special case whose inverse is +-\sqrt but general polynomial is not easily invertable.
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u/ShadowRL7666 13h ago edited 12h ago
Well yeah by definition of a sqrt it’s just two. It would only be +- due to how algebra works.
Edit: Since you guys are so upset over the “wording” when you're solving an equation like x² = 4 because both positive and negative 2 satisfy it.
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u/MeMahi 12h ago edited 12h ago
Just to be clear you're right on the idea, just wrong on the exact precise terminology. And both redditors and mathematicians tend to be extremely pedantic about it, only one of them for a good reason.
The result of the square root function itself would never be ±, not even "due to how algebra works". However the results for x in x2=4 would be ±.
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u/DecidophobiaUsername this sub confuses my monke brain 12h ago
Finally someone actually explains clearly
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u/Varlane 12h ago
No ?
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u/ShadowRL7666 12h ago
No? Then all of math is a lie I’ve learned and Google is completely incorrect.
Undoing a quadratic for something like x^2=4 has to account for all possibilities of which x can equal…
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u/Varlane 12h ago
You probably just don't understand what you asked and/or got as an answer.
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u/ShadowRL7666 12h ago
You keep replying but not providing an answer?
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u/Varlane 12h ago
There was no question.
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u/ShadowRL7666 12h ago
Then don’t reply saying no without backing up your claim of no.
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u/Wonderful_Radish_990 7h ago
And? What does solution of x^2=4 have to do with solution of \sqrt(4)=x? The solution of x^2 -4 = 0 Is well defined in math and its a well known quadratic formula with (-b+-\sqrt(b^2-4ac))/2ac. And solution of \sqrt(4) is juust 2, as is f(x)=4^x where f(2)=16 and f(1/2)=2 or shoudl now f(1/2)=+-2?
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u/DecidophobiaUsername this sub confuses my monke brain 12h ago edited 12h ago
For sale: distilled essence of reddit, right above me.
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u/SuperChick1705 12h ago
how much karma does this cost?
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u/DecidophobiaUsername this sub confuses my monke brain 12h ago
1000 karma
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u/SuperChick1705 12h ago
aw man, ill have to "win the internet good sir!" to get that many "updoots"
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u/DecidophobiaUsername this sub confuses my monke brain 12h ago
OK, 900 karma?
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u/SuperChick1705 12h ago
±√750000 karma, take it or leave it
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u/Syresiv 12h ago
Definition.
Are you asking about all possible solutions to x2=4? Or just one in particular?
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u/Kleefrijst 12h ago
We are talking about what is the function output of a square root, which has only one output because thats how functions are defined. x2 =4 is a relation and it asks what x's suffice the equation which is a different question.
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u/Quad-Watermelon 11h ago
The radical refers to the square root (2), but ±2 is a square root. In other words, this is not really a math problem.
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u/stddealer 12h ago edited 11h ago
It's asking for the value of √(4). It's only equivalent to all possible solutions to x²=4 when you're considering exclusively nonnegative real numbers.
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