Maps As Functions
In many branches of mathematics, the term is used to mean a function with a specific property of particular importance to that branch. For instance, a "map" is a continuous function in topology, a linear transformation in linear algebra, etc.
In contrast, in category theory, "map" is often used as a synonym for morphism or arrow, thus for something more general than a function.
Some authors, such as Serge Lang, use "map" as a general term for an association of an element in the range with each element in the domain, and "function" only to refer to maps in which the range is a field.
Sets of maps of special kinds are the subjects of many important theories: see for instance Lie group, mapping class group, permutation group.
In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.
A partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.
Read more about this topic: Map (mathematics)
Famous quotes containing the words maps and/or functions:
“And at least you know
That maps are of time, not place, so far as the army
Happens to be concerned—the reason being,
Is one which need not delay us.”
—Henry Reed (1914–1986)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)