# Am I right in my injective and surjective definition?

• Firepanda
In summary, an injective function maps distinct points in the domain to distinct points in the codomain, a surjective function guarantees that all elements in the codomain have at least one element from the domain that maps to them, and a bijection is a function that is both injective and surjective.

#### Firepanda

In layman terms otherwise I have trouble understanding

Injective: A function where no element on the domain is many to one.

Surjective: All the elements in the codomain have at least one element from the domain that maps to them.

I'd like to keep it simple so I can play it back to myself in the exam, rather than trying to understand a formal definition, so I hope it's ok so far..

Then

Bijection: Any map where there is no many to one elements in the Domain, but all the elements in the co-domain have at least one element mapped to them.

I'm not sure what you mean by "no element on the domain is many to one."

I would say of an injective function that "no element in the codomain is the image of more than one element of the domain," or "the function maps distinct points in the domain to distinct points in the codomain."

The definition of a surjection looks fine.

A bijection is simply a function that is both injective and surjective. The key fact here is that it has an inverse.

jbunniii said:
I'm not sure what you mean by "no element on the domain is many to one."

I would say "no element in the codomain is the image of more than one element of the domain," or "the function maps distinct points in the domain to distinct points in the codomain."

ah ok thanks, think I got a bit confused

so I can say for injections, there is at most one element in the domain that maps to one element in the codomain?

So bijections just map one distinct element in the domain to one distinct element in the codomain, such that all the elements in the codomain have an element mapped to them?

Firepanda said:
ah ok thanks, think I got a bit confused

so I can say for injections, there is at most one element in the domain that maps to one element in the codomain?

So bijections just map one distinct element in the domain to one distinct element in the codomain, such that all the elements in the codomain have an element mapped to them?

Yes, this is all correct.

yea surjection guarantees that the whole co-domain gets mapped, and injection guarantees that co-domain will be mapped by only one element.

Most simply, for f:A--> B,
injective: "if f(x)= f(y) then x= y" and
surjective: "If y is in B, there exist x in A such that f(x)= y".

## 1. What is an injective function?

An injective function, also known as a one-to-one function, is a type of mathematical function where each element in the domain maps to a unique element in the range. In other words, no two elements in the domain will map to the same element in the range.

## 2. How do I know if a function is injective?

To determine if a function is injective, you can use the horizontal line test. This involves drawing horizontal lines across the graph of the function. If no two points on the graph of the function lie on the same horizontal line, then the function is injective.

## 3. What is a surjective function?

A surjective function, also known as an onto function, is a type of mathematical function where every element in the range has at least one pre-image in the domain. In other words, every element in the range is mapped to by at least one element in the domain.

## 4. How can I tell if a function is surjective?

To determine if a function is surjective, you can use the vertical line test. This involves drawing vertical lines across the graph of the function. If every vertical line intersects the graph at least once, then the function is surjective.

## 5. Can a function be both injective and surjective?

Yes, a function can be both injective and surjective. This type of function is called a bijective function. It is a one-to-one correspondence between the domain and range, meaning each element in the domain maps to a unique element in the range and every element in the range is mapped to by at least one element in the domain.

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