Bijection - Properties

Properties

• A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.
• If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a group, the symmetric group of X, which is denoted variously by S(X), S_X, or X! (X factorial).
• Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities:

  |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:
  1. f is a bijection.
  2. f is a surjection.
  3. f is an injection.
• For a finite set S, there is 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 of elements of S is the same as the number of total orderings of that set—namely, n!.