Help with Infinity Math

Its interested.

at first, you have to review what is infinity difination?

endless, countless

you will understand where is wrong.

I haven't played with math in years, but f(1) = 0, not infinity. If you're looking for an answer to f(∞), then f(∞) = ∞, because ∞ - 1 = ∞.

Are we talking the same idea???

I gave this a huge amount of thought back in high school....the "number" in the codomain is 0 (Nought) if you follow the algebra through you will wind up with 1=2 . the explanation is due to 0 being a number of no value, it is both a negative negative and a positive negative. Oddly the reverse is true of infinity, however 0 is a real number, unlike infinity. (you can have 0 apples, you can't have infinity apples). the problem is Philosophical not mechanical.

Think hard you will work it through!

I think you're right, we're talking two different languages. I'm not sure what you mean by "codomain." Also, if this is all philosophical, then you can have infinity apples - philosophically speaking.

Interesting.

infinity is not both negative and positive. There is negative infinity and positive infinity. Zero can be defined also as being neither negative or positive simultaneously. Typically you will define positive as N >0, and negative as N<0.

See http://en.wikipedia.org/wiki/Codomain

It doesn't matter whether you define the natural numbers to be N = {1, 2, 3,......} or N = {0, 1, 2, 3,...........}

the domain of f is the natural numbers and the codomain is the set of integers. The range of f is a subset of the integers.

In other words the function f(n) = n-1 does not exist if the domain and codomain are both the set of natural numbers.

The function f(i) = i -1 where the domain and codomain are both the set of integers is surjective.

Not sure what you mean by: take f(1)=1

I'll check Wikipedia.

But I said that f(1) = 0, not 1.

OK, I read the Wikipedia entry, but I still don't understand what your confusion is. As long as x is in the set of positive integers, then everything should work out OK. It seems all you have to do is stay away from even roots of negative numbers... And I see why f(x) ≠ g(x) because f(x) can have imaginary results, where g(x) cannot.

And this is somewhat reminiscent of the definition of a function, where there can be one and only one value of y for any f(x).

That's not to say that you may be way ahead of me on these concepts.

Hi Vermin,

I think we might be talking at slightly cross purposes here: my post was a response to the original post, not to you.

Randall

I just sent an email to Jorrie. He's a very math-savvy guy on this site. It may take a day or so for him to join in, but I'm hoping he can help. He's a good guy.

Regards,

vermin

Hi elika, you asked: "If f(x)=x-1, then will it be an onto or into function [where, x belongs to N, and we take f(1)=1]?"

As has been pointed out by others before, f(x)=x-1 and f(1)=1 do not belong to the same domain.

Anyway, my feeling is that your 'confusion' arises because you want to treat infinity as a number, while it is not, as vermin has pointed out. Your "... although N goes up to infinity..." is not quite valid; I prefer to say that "N approaches infinity", which is more correct and is just another way of saying N is very large, without limit.

-J

Nothing complicated, I guess.

f(x)=x-1, when x=1 then x-1=0

when x=2, then x-1=1

when x=0, then x-1=-1

when x=N, then x-1=N-1, N can be + or – number

when x=∞, then x-1=∞-1, which is still same as ∞ (INFINITY!)

Jorrie is correct, it is always better to think in terms of approaching a limit of infinity (or negative infinity).

The brute force way to see how different values of x effect f(x). Start with

x=1, f(1)=0

x=10, f(10)=9

x=100, f(100)=99

x=1000, f(1000)=999

x=10000, f(10000)=9999

as x approaches +∞, f(x) approaches +∞. Although you can say x approaches +∞ faster then f(x) approaches +∞, although it is only by the value of 1.

x=-1, f(-1)=-2

x=-10, f(-10)=-11

x=-100, f(-100)=-101

as x approaches -∞, f(x) approaches -∞. You can say f(x) approaches -∞ faster than x approaches -∞.

You might want to study up on limits in series in mathematics. I am weak in math but this is begining calculus course work, get familiar with Limits of functions. And isnt this statement wrong? f(x)=x-1, when x=1 f(1)=1. f(1)=0 ,f(1)≠1. f(2)=1 instead. Or am i just completely missing something?

I'm going to have another go at trying to explain this:- you say

"there would remain 'one' no. in the codomain, which wouldn't correspond to any in the domain."

Suppose there is 'one' no. call it n then the number in the domain which corresponds to it is n+1. CONTRADICTION.

It's also easy to define a surjective (onto) function from one set to another where all your initial instincts tell you that the codomain is much bigger than the domain. Look at this function from the Natural numbers ( {1, 2, 3, 4, ........} ) to the set of Integers ( {0, 1, -1, 2, -2, 3, -3, ...........} )

1 -> 0

2 -> 1

3 -> -1

4 -> 2

if n is even: then i = n/2 else i = -[(n-1)/2)]

Again it's easy for me to tell you which number in N corresponds to any number in I.

This is because the integers are "countably" infinite.

Even the rational number (the set Q) are countably infinite. If you look on Wikipedia and elsewhere on the net you can easily find surjective functions from N to Q

The actual question was:-

Show that the function 'f':N→N defined as f(x)=x-1, for every x>2, is not one-one but onto.

condition- f(1)=f(2)=1.

it isn't one-one is clear to me, but why it isn't onto, i don't seem to understand, n that's because,

no.of natural no's is infinite, but even then, ∞-1 no's only form the range, while there are actually ∞ no's in the codomain.

& why do we take ∞-1=∞ (if that solves the problem)?

can we then say that

∞²=∞?

no.of odd no's=no.of even no's=no.of natural no's?

to me the, 'infinity' is somewhat similar to zero, coz if we add, subtract, multiply or divide anything the result is the same!

{except: ∞-∞=0, ∞/∞=1}

I think you mean:

Define: f(x) = x-1
What is: x if f(x)=x ?

If we are working in the linear limit domain, I don't believe this equation has a proper solution, even though we generally say that infinity=infinity-1.

Clarification, please.

that's wat i'm arguing abt!!

that, though we say ∞-1=∞, here it doesn't hold true,

perhaps, coz this is mathematics, and approximations are better used in physics.

Inherent in the concept of infinity as mathematicians hold it infinity is an impossibly big number it means "like, everything." And infinity is so absolute that ∞-1 = ∞.

There's an old joke about math geeks. They're on a bus, signing the x number of beers on the wall song, but they're singing, "Infinity number of beers on the wall. Infinity numbers of beers. Take one down and pass it around, infinity numbers of beers on the wall."

Infinity is mathematically speaking the concept of everything, so subtracting one from everything is "sort of like" trying to divide by zero... It's undefined. How can you take one away from the set of everything? By it's very definition, it's the set of everything.

Sorry Vermin, infinity does not mean "like everything". There are many different infinities. The set of integers is the smallest such. The set of all rationals is the same size - but that does not mean that the sets are identical - that is like saying that three apples is identical to three oranges. The size of a set is best defined in terms of other sets to which you can create 1-to-1 correspondences. The next smallest well-identified infinity after the integers is the set of all points on a line. When I studied maths (over half a century ago) it was not known whether there were any infinities of intermediate size - I don't know whether that is still true.

A common feature to all infinities is that an infinite set contains subsets of the same size, and that the number of such subsets is an infinity that is greater than the parent set. So, no matter how large your infinite set, you can always define one that is larger (i.e. there are only many->one correspondences). That makes the concept of "everything" in that particular mathematical sense to be somewhat of a self-contradiction.

(Same guest)

Yes, the statement still holds true. The argument is about infinity being "the solution" to this equation is not that infinity fails to satisfy the equation. It is that there are many different infinities that can satisfy the equation, so to say "infinity is THE solution" is at best incomplete - the correct statement would be "the solutions are (all) infinite".

In addition, there are mathematical systems in which it is more correct to say that "there is no solution" than that "the solutions are infinite". This depends on how you define "=". (It pays to remember that the meaning of mathematics depends very critically on definitions)