Bijection, Injection, and Surjection

[amcavoy]

I was just looking at the definitions of these words, and it reminded me of some things from linear algebra. I was just wondering: Is a bijection the same as an isomorphism?

 

[Jonny_trigonometry]

funny, 'cause I was wondering what a bijection was earlier today while riding the train. I can't really follow the line of thought in the book I'm reading without a good idea of what a bijection is.

 

[Icebreaker]

A mapping f:A->B is injective if it's one-to-one. But elements of B may be unmapped. f is surjective if every element of B is mapped from some a in A by f(a). f is bijective if it's surjective and injective simultaneously.

It's not an isomorphism because an isomorphism is a function between two rings that preserves the binary operations of those rings, on top of which the function is bijective.

 

[DeadWolfe]

A mapping f:A->B is injective if it's one-to-one. But elements of B may be unmapped. f is surjective if every element of B is mapped from some a in A by f(a). f is bijective if it's surjective and injective simultaneously.
It's not an isomorphism because an isomorphism is a function between two rings that preserves the binary operations of those rings, on top of which the function is bijective.

I think that an isomorphism can preserve any structure, so an isomorphism between groups preserves the group operation, isomorphisms between rings preserves ring operations, isomorphisms between vector spaces preserves scalar multiplication and vector addition, etc...

I could be wrong though.

 

[Hurkyl]

As the others have mentioned, the definition of "isomorphism" depends on the objects you're manipulating. Incidentally, a bijection is an isomorphism of sets.

An isomorphism is defined to be a homomorphism with an inverse that is also a homomorphism.

A homomorphism of sets is just a function.
A homomorphism of topological spaces is just a continuous function.
A homomorphism of vector spaces is a linear transformation.

 

[TOKAMAK]

So would a bijective homomorphism be an isomorphism? I was taught (working with vector spaces) that a linear bijection is an isomorphism. Are these okay?

 

[matt grime]

You're not too far off, tokomak; strictly speaking an *invertible* homomorphism is an isomorphism, but if it helps you can use the bijective idea instead as it will suffice for most things you'll meet in undergraduate mathematics.

 

[jcsd]

You're not too far off, tokomak; strictly speaking an *invertible* homomorphism is an isomorphism, but if it helps you can use the bijective idea instead as it will suffice for most things you'll meet in undergraduate mathematics.

What's the difference? Are there bijective homomorphisms which are not invertible?

 

[DeadWolfe]

No, but there are invertible homomorphisms which are not bijective.

 

[Hurkyl]

Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

 

[DeadWolfe]

Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

Well, for a mapping to be an isomorphism, is it required that it's inverse be a homomorphism?

 

[Muzza]

No, but there are invertible homomorphisms which are not bijective.

I could've sworn that a function (homomorphism or not) was invertible iff it was bijective.

 

[Hurkyl]

Well, for a mapping to be an isomorphism, is it required that it's inverse be a homomorphism?

Yes! In some cases, the inverse is automatically a homomorphism, but not in all. For example, consider the identity map \bar{\mathbb{R}} \rightarrow \mathbb{R} where I am using \bar{\mathbb{R}} to denote the discrete topology. This is clearly an invertible map, but it is not a homeomorphism. (Which is what we call isomorphisms when doing topology)

 

[jcsd]

Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

That does make more sense, looking up the catergory theory definition of an isomorphism, a morphism isn't an isomorphism unless it is invertible and it's inverse is a morphism.

Still, the way that I informally think about homomorphisms, it is hard to imagine a bijective homomorphism whose inverse is not also a homomorphism. I'd guess that this is probably because in all the catergories which form my view of isomorphisms all bijective homomorphisms are isomorphisms.

Can you give me an example of a catergory with bijective homomorphisms which are not isomorphisms?

 

[matt grime]

For jcsd:

Hurkyl just did: the identity mapping from any set with the discrete topology to itself with some other topology.

The map of differential manifolds from [0,1] to itself x-->2^2 is not invertible in the space of differential manifolds with diffeomorphisms (the inverse has no tangent at 0).

For Muzza: There are also plenty of isomorphisms in categories where it does not even make sense to start talking about bijections since the morphisms in no meaningful way act on elements of a set: morphisms are just arrows, they do not have to be maps on any underlying sets. This is one distinction between category theory and set theory.