Bijections result when the function is surjective and injective

[laminatedevildoll]

Bijections result when the function is surjective and injective.

How do I find a bijection in N and the set of all odd numbers?

f(x) = 2x+1

Do I have to prove that this is one-to-one and onto? Am I on the right track?

 

[honestrosewater]

Bijections result when the function is surjective and injective.

How do I find a bijection in N and the set of all odd numbers?

f(x) = 2x+1

Do I have to prove that this is one-to-one and onto? Am I on the right track?

Yes. Can you prove that for all n and m in N, if f(n) = f(m), then n = m? (Just plug and play.)

 

[HallsofIvy]

You would also, of course, have to prove that it is a surjection: that is, that if m is an odd number then there exist an integer n such that 2n+ 1= m.

 

[BicycleTree]

You can get the surjection from the definition of odd number.

 

[laminatedevildoll]

Thank you for your help. I think that I need a lot of practice using the definition.

 

[honestrosewater]

Which definition- bijection?

 

[laminatedevildoll]

Which definition- bijection?

Yeah. It's kind of frustrating to prove if a function is one-to-one or onto when one way of proving for one problem doesn't apply to the next. It's all about playing around with the definition and all.

 

[laminatedevildoll]

What's the difference between finding a bijection between P(N) and P(Z) AND N and Z? P stands for power set (the curly P).

 

[BicycleTree]

You mean a bijection between P(N) and P(Z) compared to a bijection between N and Z?

I've never done that kind of thing but I think if you have a bijection between N and Z then you automatically have a bijection between P(N) and P(Z) by replacing every element x of N in a given element y of P(N) with f(x) and mapping y to the resulting element z, which will be in P(Z). If you have a bijiection between P(N) and P(Z) then I don't think you necessarily can say you have a bijection between N and Z.