Application

Application

An application is a relation of a set E toward a set F so that all elements in set E have one and only one image in set F.

An application of a set E in itself is called a transformation of set E.

Synonym for function.

Properties

  • Bijective application
    Application that is both injective and surjective.
    Synonym for bijection.
  • Injective application
    Application f of a set E toward a set F in which the distinct elements of the domain have distinct images.
    Synonym for injection.
  • Surjective application
    Application f of a set E toward a set F in which the image is equal to the set of arrival F.
    Synonym for surjection.

Examples

  • The relation of \(\mathbb {N}\) in \(\mathbb {N}\) under which every whole number x is made to correspond to its double 2x is an injective application of \(\mathbb {N}\) in \(\mathbb {N}\), also known as an injection.
  • The relation of \(\mathbb {Q}\) in \(\mathbb {Q}\) under which every rational number x is made to correspond to its half 0.5x is a bijective application of \(\mathbb {Q}\) in \(\mathbb {Q}\), also known as a bijection, because on the one hand, each rational number corresponds to its half, and on the other hand, each rational number is the half of another rational number.
  • The relation of \(\mathbb {Z}\) in \(\mathbb {N}\) under which every integer x is made to correspond to its absolute value |x| is a surjective application of \(\mathbb {Z}\) in \(\mathbb {N}\), also known as a surjection, because every whole number is the image (the absolute value) of at least one integer.

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