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In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f ( x ) = x is true for all values of x to which f can be applied.
When s ≠ 0, this is an example of a quasiconformal homeomorphism that is not smooth. If s = 0, this is simply the identity map. A homeomorphism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal.
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 ...
The identity map on any module is a linear operator. For real numbers, ... As a simple example, consider the map f: R 2 → R 2, given by f(x, y) = (0, y).
Identity For every object X, there exists a morphism id X : X → X called the identity morphism on X, such that for every morphism f : A → B we have id B ∘ f = f = f ∘ id A. Associativity h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever all the compositions are defined, i.e. when the target of f is the source of g, and the target of g is the ...
For example: A semigroup homomorphism is a map between semigroups that preserves the semigroup operation. A monoid homomorphism is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a 0-ary operation).
the inclusion, a retraction is a continuous map r such that =, that is, the composition of r with the inclusion is the identity of A. Note that, by definition, a retraction maps X onto A. A subspace A is called a retract of X if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (any constant map ...
For example, a map from the unit circle to any space is null-homotopic precisely when it can be continuously extended to a map from the unit disk to that agrees with on the boundary. It follows from these definitions that a space X {\displaystyle X} is contractible if and only if the identity map from X {\displaystyle X} to itself—which is ...