# Bijections result when the function is surjective and injective

• laminatedevildoll
In summary, a bijection is a function that is both surjective and injective. To find a bijection between N and the set of all odd numbers, you can use the function f(x) = 2x+1. To prove that f(x) is a bijection, you need to show that it is both one-to-one and onto. This can be done by plugging in values for n and m in N and showing that f(n) = f(m) implies n = m, and also by showing that for any odd number m, there exists an integer n in N such that 2n+1 = m. The definition of odd numbers can help with this proof. When trying to find a bijection
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?

laminatedevildoll said:
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.)

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.

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

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

Which definition- bijection?

honestrosewater said:
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.

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).

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.

## 1. What is a bijection?

A bijection is a type of function in mathematics that has two important properties: surjectivity and injectivity. This means that every element in the function's output has a unique input, and every element in the function's input has a corresponding output.

## 2. How do you know if a function is a bijection?

A function is a bijection if it is both surjective and injective. In other words, every input has a unique output and every output has a corresponding input. If these two conditions are met, then the function is a bijection.

## 3. What is the difference between a surjection and an injection?

A surjection is a function where every element in the output has a corresponding input. An injection is a function where every element in the input has a unique output. A bijection is a function that has both of these properties.

## 4. Can a function be a bijection if it is not surjective or injective?

No, a function must be both surjective and injective to be considered a bijection. If a function is not surjective, it means that there are elements in the output that do not have a corresponding input. If a function is not injective, it means that there are elements in the input that have more than one corresponding output.

## 5. What is the importance of bijections in mathematics?

Bijections are important in mathematics because they allow us to create one-to-one correspondences between two sets. This can be useful in solving problems and proving theorems. Additionally, bijections can help us understand the relationships between different mathematical objects and structures.

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