Bijection Totally Explained

Everything about Bijection totally explained

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there's exactly one x in X such that
f(x) = y.
Alternatively, f is bijective if it's a one-to-one correspondence between those sets; for example, both one-to-one (injective) and onto (surjective). (One-to-one function means one-to-one correspondence (for example, bijection) to some authors, but injection to others.)
For example, consider the function succ, defined from the set of integers





, that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x + y, x − y).
A bijective function from a set to itself is also called a permutation.
The set of all bijections from X to Y is denoted as X

o [-1,1] : x mapsto sin(x)

isn't a bijection because π/3 and 2π/3 are both in the domain and both map to (√3)/2.

Properties

A function f from the real line R to R is bijective if and only if its plot is intersected by any horizontal line at exactly one point.

If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (^o), form a group, the symmetric group of X, which is denoted variously by S(X), S_X, or X! (the last reads "X factorial").

For a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities: ť |f(A)| = |A| and |f⁻¹(B)| = |B|.

If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent: ť # f is a bijection.

# f is a surjection. ť # f is an injection.

At least for a finite set S, there's a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations (another name for bijections) of elements of S is the same as the number of total orderings of that set -- namely, n!.

Bijections and category theory

Formally, bijections are precisely the isomorphisms in the category Set of sets and functions. However, the bijections are not always the isomorphisms. For example, in the category Top of topological spaces and continuous functions, the isomorphisms must be homeomorphisms in addition to being bijections.

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.

Everything about Bijection totally explained

Properties

Bijections and category theory

See also