Recursive Definitions

ok, I feel stupid. I'd rather ask though..

Page 14 of the study guide: Sequences:

to me the recursive definition of a sequence should be the same as the definitions on the previous pages. let me explain:

EVEN is defined as: (2,4,6,8,...)
Universal set R
generator 2
function: f(x) = x+2

For the sequence (-1,2,5,3n-4)
Now my first thought was:
Universal Set Z (integers)
generator 1
function f(x) = 3x-4

However, obviously 1 isnt in the sequence, so it can tbe a generator. -1 is in the sequence though, and is seems to me I must now find the function that gets me to the next number in the sequence, so that I could use it in the recursive definition. Now the study guide tells me 1) I need a universal set of ZxZ (ie. ordered pairs). I dont understand why. I am thinking tht this is to specify the inputs to the function f as such (-1,1)(2,2)(3,5)(n,f(n)).. right? the next pair in this sequence should be (n+1,f(n)+3)

This simply concludes that F(N+1) would give F(n) + 3 (ie. The function with the next N would give the answer of function of the previous N + 3.

I talking a lot - perhaps the question here is: Are we using ordered pairs and the universal set of ZxZ ONLY because we need to explain the recursive definition in terms of functions with input/output? (because of not having a more simple generator and function)(in stead of having a single generator, now we have F(1) = -1 as the generator?)
Im afraid of misunderstanding something here..

Try this sum: 3(1)-4 = ?

I think that might be all you need to get you on your way.

-1

I think I got to understand it better somewhere last night around 12. My math is either outdated or something.
Think my question is why is the domain of the function Z++xZ++ for the sequence and only Z++ for the function f(x) = x+ 2 of the set of INTEGERS?

Danie van Eeden
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Hmm, now you've got me thinking ... OK on the fly then ...

You're trying to build whatever starting with your generator (or as few of these as possible). So then your generator has to be a member of whatever it is you're defining. ie. if it's a kind of number you're defining, then your generator has to be a number at very least. Ordered pairs don't make very nice numbers.

The generator has to be a number, but all your function has to do is take ... well it has to take at least the generator doesn't it? ...

um... so the function has to be from {kind of thing the set is made of} --> {kind of thing the set is made of}

Looks like the function shouldn't be from ordered pairs to numbers then... <<scratch head at this point and look somewhat puzzled>> ... I'll go and compare to what you've set out symbolically then...

OK. Yes if you look at part 2 of "the short version", I think you'll find that we have a rough "answer" to at least that.

But part 1??? ... OK ... There you've got me stumped for the time being.

What would a function to form a sequence require as input? Depends, I suppose. It would probably at least need the first term (or a way to produce this). ... um .... is it from pairs of (This_Item, Next_Item) to Next_Item?

Where you have a set of all possible pairs (This, Next) to choose from?

Sounds a bit deadendish to me... Like I said, you've got me stumped. I'm just drawing a little map of my dead end, so you can see at least one of the places one can get stuck in.

Hi anf thx for the reply. I think i figured it just as I read your reply.

The normal sequence 2,4,6,8 is easy. you can immediately see 2 is a generator and part of Z++.
In the second example, all they did was define the generator in terms of the function f(x) ie. the generator = 3n-4 and then 3(1) -4, This means that the generator isn;t defined as a number from Z++ but as a function from Z++>z++. (becuase function has input variable and output variable ; hence (1;-1) why do they do this?? stil trying to figure. perhaps because they are unable to get from -1 to 2 to 5 to 8 next without it and maybe becuase they are trying to force us to use 3n-4 function. Im just trying to figure why we cant define -1 as generator and simply say fx = x+3; with universal set being Z (integers). when I figure out that, I'll be getting somewhere. any thoughs? (at least I think ive got the Z++->Z++)

Danie van Eeden
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No. The "generator" is some constant, so the formula associated with the function (which if we're picky, we take to mean the Actual Mapping - not the rule that produces it) can't be the generator. Yes, it "generates" things. It's the input transformation machine. No that doesn't make it a generator. For some strange reason, 2 is the "generator" of the first mapping, not 2n+1, for instance.

The generator for the second function is 1. This number, 1 , can then be fed into the formula for one of the functions that map something like numbers to further members of the set.

Look, I think you could define -1 as a generator (as long as you change your Universal Set to the whole of "Tsett" - as I prefer to call it). They're just letting you see things defined in terms of a different formula (which formula spits out image values). Take the ordered sequence 1 ... Inf, and your formula will produce a mapping from these to all numbers of the kind you seek.

The formula is a much better candidate for being viewed as "an actual generator" (because it "generates" - it pumps out numbers that are in the set -- and every one of these, too), but FORMULA is NOT termed the "generator".