# Newbie Asks: ONTO Surjection Help

• xfunctionx
In summary, the conversation discusses the concept of surjective maps and how they are proven to be surjective. It is mentioned that for a linear map, it is easier to deal with a basis and show that it is in the image set. The problem linked to in the conversation demonstrates this concept, as (1,1) and (1,2) are shown to be in the range and are asked if they form a basis for R^2. The conversation ends with the original poster expressing their gratitude for the explanation.

#### xfunctionx

Hi I'm new on these forums, and not only is this my first post, but this is also my first thread.

The following is not a homework question, but a question I found. However, I have no idea how to do this. I would appreciate it if someone could help me. Please click on the following link :tongue:

Welcome to the forums! To show in general that a map T is surjective you need to show that for every element in v in the range, there exists an element w in the domain such that T(w) = v. With a linear map, it is a little easier since you can deal with a basis. To show that a linear map is surjective, it is sufficient to show that a basis for the range is in the image set. In the problem you linked to, you can see that (1,1) and (1,2) are in the range. Do those two vectors form a basis for R^2? If so, the map is surjective.

eok20 said:
Welcome to the forums! To show in general that a map T is surjective you need to show that for every element in v in the range, there exists an element w in the domain such that T(w) = v. With a linear map, it is a little easier since you can deal with a basis. To show that a linear map is surjective, it is sufficient to show that a basis for the range is in the image set. In the problem you linked to, you can see that (1,1) and (1,2) are in the range. Do those two vectors form a basis for R^2? If so, the map is surjective.

Thank you very much eok20. I think I finally understand

## 1. What is ONTO Surjection?

ONTO Surjection is a mathematical function that maps elements from one set to another, where every element in the output set has at least one corresponding element in the input set. In simpler terms, it means that no element in the output set is left without a match from the input set.

## 2. How is ONTO Surjection different from other types of functions?

ONTO Surjection is different from other functions because it guarantees that every element in the output set has a corresponding element in the input set. Other functions, such as ONTO Injection, do not have this requirement and may result in some elements in the output set having no matches from the input set.

## 3. Can you provide an example of ONTO Surjection?

One example of ONTO Surjection is a function that maps the set of all real numbers to the set of all positive real numbers. Every positive real number has a corresponding element in the set of all real numbers, making it an ONTO Surjection.

## 4. What is the importance of ONTO Surjection?

ONTO Surjection is important because it ensures that the output set is fully utilized and that no element is left without a match. This is useful in various fields such as computer science, economics, and statistics, where data needs to be accurately mapped and analyzed.

## 5. How is ONTO Surjection represented mathematically?

ONTO Surjection can be represented using set notation, where the function is denoted as f: A → B, where A is the input set and B is the output set. It can also be represented using a graph, where every element in the output set has at least one arrow pointing to it from the input set.