Function Notation Homework: Surjective but Not Injective

In summary, the conversation discusses the creation of a function that is surjective but not injective, with the use of notation and words to explain the function.
  • #1
ballzac
104
0

Homework Statement


Give an example of a map from the set N of positive integers to itself which
is surjective but not injective.


Homework Equations





The Attempt at a Solution


It's easy to come up with an example, but I'm not sure on notation.
Here's how I've written it, but I know it's not quite right. I'm sure you can see the function that I am meaning to give. Any help with the notation would be appreciated. If it's not clear what function I mean, let me know and I will put it into words :)

[tex]\lbrace f:f(1)=1, f(s)=s-1,s>1\rbrace[/tex]
 
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  • #2
That's plenty clear enough. Just defining f(1)=1 and f(s)=s-1 for s>1 without using the {} notation is also fine.
 
  • #3
Oh cool. Thank you for your quick response. I may actually just define the function as you say and use words to explain it if it comes up in the exam. I think the lecturer prefers things explained as much in words as possible anyway.
 
  • #4
That's plenty clear enough. Just defining f(1)=1 and f(s)=s-1 for s>1 without using the {} notation is also fine.
 

1. What is function notation?

Function notation is a way of representing a mathematical function. It is typically written as f(x), where "f" is the name of the function and "x" is the input or independent variable.

2. What does it mean for a function to be surjective but not injective?

A surjective function, also known as onto, means that every element in the range of the function has at least one corresponding element in the domain. On the other hand, an injective function, also known as one-to-one, means that each element in the range of the function has only one corresponding element in the domain. So, a function can be surjective but not injective if there are elements in the range that have more than one corresponding element in the domain.

3. How can you tell if a function is surjective but not injective?

To determine if a function is surjective but not injective, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not injective. To check if the function is surjective, you can check if every element in the range has at least one corresponding element in the domain.

4. What is an example of a surjective but not injective function?

An example of a surjective but not injective function is f(x) = x², where the domain and range are both real numbers. Every element in the range has at least one corresponding element in the domain, but there are elements in the range that have more than one corresponding element in the domain (e.g. f(2) = f(-2) = 4).

5. Why is it important to understand the difference between surjective and injective functions?

Understanding the difference between surjective and injective functions is important because it helps us understand the behavior and properties of different types of functions. For example, surjective functions are often used in modeling real-world situations, while injective functions are important in fields such as cryptography and data encryption. Additionally, knowing the difference between these two types of functions can help in solving various mathematical problems and proofs.

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