Math Is Fun Forum

Kazy

Member

Registered: 2006-01-24

Posts: 37

Bijection

I need to prove the following:

a) Prove that the function f(n)={  2n, if n>0  and -2n+1, if n ≤ 0} is bijective.
b) Prove that Z ≈ Z+ by finding a bijective function g: Z+ -> Z.
c) Let Z- be the set of negative integers. Prove that Z- ≈ Z+ by finding a bijective function f: Z+ -> Z=. prove that your function is bijective.

I know to prove something is bijective you need to show its surjective and injective, however I'm completely lost from there. Any help?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Bijection

For a, take a step back.  Look at what it is saying.  Don't see 2n and don't see -2n+1.  See odd and even.  Now it should become clear, that all that the map is a bijection since all evens and all odds are hit, and only positives go to evens and only negatives go to odds.

For b, lets do something almost exactly like a.  Lets have the function f where f(x) = {x/2 if x is even and -(x-1)/2 if x is odd}.

C is probably the easiest one of all.  f(x) = -x.

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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..."