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amcavoy
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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?
Icebreaker said: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.
matt grime said: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.
Hurkyl said:Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!
No, but there are invertible homomorphisms which are not bijective.
Yes! In some cases, the inverse is automatically a homomorphism, but not in all. For example, consider the identity map [itex]\bar{\mathbb{R}} \rightarrow \mathbb{R}[/itex] where I am using [itex]\bar{\mathbb{R}}[/itex] 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)Well, for a mapping to be an isomorphism, is it required that it's inverse be a homomorphism?
Hurkyl said:Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!
Bijection, injection, and surjection are all types of functions in mathematics. A bijection is a function that is both injective (one-to-one) and surjective (onto). An injection is a function that is injective but not necessarily surjective. A surjection is a function that is surjective but not necessarily injective.
A function is a bijection if and only if it is both injective and surjective. To determine if a function is injective, you can use the horizontal line test - if every horizontal line intersects the graph of the function at most once, then the function is injective. To determine if a function is surjective, you can use the vertical line test - if every vertical line intersects the graph of the function at least once, then the function is surjective.
Bijection, injection, and surjection are important concepts in mathematics because they help us understand the properties of functions and how they relate to one another. They also have practical applications in various fields, such as computer science and economics, where these types of functions are used to model real-world scenarios.
Yes, a function can be both an injection and a surjection. In fact, a bijection is both an injection and a surjection, as it is a function that is both one-to-one and onto.
Bijection, injection, and surjection are all related to inverse functions. A bijection has an inverse function, which is also a bijection. An injection has a left inverse, while a surjection has a right inverse. In general, a function must be a bijection in order for it to have an inverse function.