Math Is Fun Forum

EPhillips1989

Member

Registered: 2007-11-03

Posts: 29

surjective function

hey can anyone ive me an example of a surjective function when
Z*N->Q???

will be much appreciated if you could share your ideas thanks in advance xxx

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: surjective function

What does Z*N->Q mean?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: surjective function

Assuming the function is Z and it goes from the natural to the rationals, there is an explicit formula you can find.  You could probably find it through google.  But there is another way you can do it.  We let:

{(a, b) : a, b are in N, and gcd(a, b) = 1}

Represent our positive rational numbers.  We then order this set by the relation:

(a, b) < (c, d) iff a < c or a = c and b < d

We notice that based on this definition of order, there is certainly nothing less than (1, 1), and that this is a total ordering in our set.  Now we map:

1 -> 1st element of this set
2 -> Negative of the first element of this set
....
...
2n -> nth element of this set
2n+1 -> Negative of the nth element of this set

And the claim is that such a mapping is surjective.

Edited to add: You also need to stick 0 in the beginning.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: surjective function

I'm guessing that Z*N means

.

If that's the case, then you can just take a∈Z and b∈N and define a rational a/b.

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Why did the vector cross the road?
It wanted to be normal.