# Map

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In mathematics and related technical fields, the term map or mapping is often a synonym for function. Along these lines, a partial map is a partial function, and a total map is a total function.

In many specific branches of mathematics, the term is used for a function with a specific property relevant to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc.

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.

A mapping m which has domain A and codomain B can be denoted symbolically as

${\displaystyle m:A\rightarrow B\ }$.

If element a belongs to the domain A and if element b belongs to the codomain B and if mapping m maps element a to element b, this can be denoted symbolically as

${\displaystyle m:a\mapsto b}$,

"m maps a to b", which means the same as m(a) = b, "function m of a is b", in function notation. In such case, element b can be referred to as the image of element a.

For any element a in a mapping's domain, a has one and only one image. That is, there exists a b such that ${\displaystyle m:a\mapsto b}$ and if ${\displaystyle m:a\mapsto c}$ then b = c.

A mapping m is an injective mapping m : AB if each element of A is mapped to a different element of B (note the use of arrow with hook). Injective mappings have the property that the domain's cardinality is less than or equal to the codomain's cardinality: card A ≤ card B.

A mapping m is a surjective mapping m : A B if every element of B is an image of some element of A (note the use of two headed arrow): that is, if the mapping's range is equal to the mapping's codomain. Surjective mappings have the property that the domain's cardinality is greater than or equal to the codomain's cardinality: card A ≥ card B.

A bijective mapping is both injective and surjective, and has the property that the domain's cardinality equals the codomain's cardinality. A mapping has an inverse mapping if and only if the mapping is bijective. Why? If the mapping is bijective, then it is evident that it has an inverse. Now, suppose that a mapping ${\displaystyle m:A\rightarrow B}$ has an inverse ${\displaystyle m^{-1}:B\rightarrow A}$. This inverse can only be a mapping if every element of its domain B has an image, which can only be the case if mapping m is surjective. Also, the inverse can be a mapping only if every element of its domain has no more than one image, which can be the case only if mapping m is injective. So if mapping m has an inverse mapping, then m must be bijective.

A mapping's domain and codomain are uniquely defined, so that if m : AB is some mapping, and if n : BC is an inclusion mapping, C being a proper superset of B, then the mapping n o m which is their composite is a different mapping from m.