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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).
A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1]
Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets. Closed function: maps closed sets to closed sets. Compactly supported function: vanishes ...
For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function , that maps the real numbers onto the positive numbers.
The set of all bijective functions f: X → X (called permutations) forms a group with respect to function composition. This is the symmetric group , also sometimes called the composition group . In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the ...
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
Variations of the horizontal line test can be used to determine whether a function is surjective or bijective: The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at least once. f is bijective if and only if any horizontal line will intersect the graph exactly once.
For example, R is an ordering ≤ and S an ordering , then an isomorphism from X to Y is a bijective function : such that (). Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism .