Comments on: classes of functions

By: Dar

Dar — Sun, 04 Sep 2011 04:23:35 +0000

thanks, this helped

By: Information Overload 2011-04-24 « citizen428.blog()

Information Overload 2011-04-24 « citizen428.blog() — Sun, 24 Apr 2011 17:33:07 +0000

[...] classes of functionsIf you often get confused by terms like injective, surjective and bijective, this blog post including diagrams and programming analogies should help. [...]

By: Josh

Josh — Tue, 19 Apr 2011 05:39:54 +0000

Just gonna chime in here, it’s usually a “partial relation” and not a partial function, since “function” has a very specific meaning, i.e. each item in the left is mapped to AT MOST one item on the right. Additionally, an injective function does not necessarily mean every input has an output. It means that, for those input that DO have an output, that output is unique. What you’re describing is a total, injective function.

By: chepprey

chepprey — Mon, 18 Apr 2011 13:52:04 +0000

Would you mind sharing with us what your purposes are? Because, for the purposes of the branch of mathematics where these classifications were devised, there obviously IS an important difference.

By: Horia

Horia — Mon, 18 Apr 2011 13:51:28 +0000

Surjective functions find a lot of use as hash functions for hash tables and their variants. Ideally, you’d want every bucket in a table to have some objects assigned to it, therefore the hash function must be designed such that it is surjective.

Also, from a mathematical point of view, partial functions aren’t really functions. By definition, a function must map each object in its domain (persons here) to an object in the codomain (bikes). Real life is more interesting though

By: Sid

Sid — Mon, 18 Apr 2011 13:47:17 +0000

In general, a bijection is defined as a 1-1 onto map, while an isomorphism is defined in abstract algebra as a bijective map that preserves structure. Since sets have no structure, set isomorphisms are indeed bijections. For objects with any sort of additional structure (well-ordered sets, posets, groups, rings, fields, lattices, etc) though, they aren’t the same.

By: hdevalence

hdevalence — Mon, 18 Apr 2011 13:44:14 +0000

Well, a map being a bijection is a property that just has to do with sets, whereas a map being an isomorphism is a stronger property: it means that the map is a bijection which also preserves some extra mathematical structure.

Bijections preserve cardinality (size), while isomorphisms preserve structure.

If you only consider things as sets, there is no extra structure to preserve, so any bijection is an isomorphism. But if you consider things with additional mathematical structure, the two are no longer equivalent. For example, it is possible to find a bijection between the real line and the plane. So, considered as sets, they are isomorphic. But considering them as vector spaces, where we can add vectors and scale them, they are not isomorphic, because the real line is one-dimensional and the plane is two-dimensional. Whatever bijection we find between them will not preserve the vector space structure, which is important.

By: Brandon

Brandon — Mon, 18 Apr 2011 12:49:49 +0000

In the category of sets bijective is the same as isomorphism. If there is no further structure on the sets (as is the case with people and bikes), then it doesn’t matter which term you use.

By: numerodix

numerodix — Sun, 24 May 2009 11:18:52 +0000

I figured someone would bring this up, I know they are not exactly the same but similar. I have no idea what a topological space is, though.

Anyway, for my purposes they are the same, thus far.

By: chithanh

chithanh — Sun, 24 May 2009 10:23:29 +0000

Small nitpick: bijective is not the same as isomorphic. For example, in topological spaces, a bijective continuous function is only an isomorphism if the inverse is also continuous.