# Am I right in my injective and surjective definition?

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.

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org
jbunniii
Science Advisor
Homework Helper
Gold Member
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.

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?

jbunniii
Science Advisor
Homework Helper
Gold Member
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.

HallsofIvy
Science Advisor
Homework Helper
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".