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Function

Ordered pairs from the corresponding elements of the two ordered sets can also be well defined. In other words, a particular element of another set can be assigned to each element of a set. This assignment is the basic concept of a function. If the rules of assignement  is well defined, the ordered pairs become more useful. For example, let X and Y be nonempty sets. A function ƒ of mapping from X to Y is an assignment of only one unique element y in Y to each element  x in X. Since the assignment is unique, the function can be expressed as ƒ(x)=y. And the assignment or mapping can be expressed as ƒ:X→Y.

Similar to set, a function can be specified in many different ways. The most common way is the using of a formula to define a function. But a function usually specific a relation from X to Y and is just as a subset of X⨯Y. And the relation can only relate one, and only one element y from Y for every element x in X to form an unique ordered pair. For example, a relation R between X and Y is a subset of X⨯Y, since x∈X, y∈Y, and (x,y)∈R,  therefore R forms a subset of X⨯Y and can be expressed as xRy for expressing x is related to y by therelation R.

Specific names are used to name the two sets of elements for a function. Let function ƒ maps X to Y.  For a function ƒ from X to Y, set X is the domain of ƒ and set Y is the codomain of ƒ. And for the relation ƒ(x)=y, element y is the image of element x and element x is the preimage of element y. Since all elements of Y are not necessary to be the image of element x, the set of all images of elements of X is called the range, or image of  function ƒ.

Special Types of Functions

When the range of function ƒ mapping from X to Y is equal to the whole codemain Y, the function ƒ is said to be a function from X onto Y. or an onto function or a surjection from X to Y.

When the outputs of a function ƒ produced  the corresponding inputs are always different to each other for all inputs of function ƒ, that is if i,j∈X and i≠j then ƒ(i)≠ƒ(j) for any i, j, the function ƒ is said to be a one-to-one (1-1) function, or an injection from X to Y.

When a function is both a one-to-one and onto function, that is all elements of Y are the image of element x and all images of elements of X are different to each other, the function is called one-to-one onto function or a bijection between X and Y. This is a perfect one-to-one correspondence between the elements of the two sets X and Y. Since no image of elements of X is repeated is Y, a bijection can only exist iff X and Y have the same number of elements.

Characters of a Function

A function ƒ from X into Y,  ƒ:X→Y, ƒ is also a relation from a set X to a set Y. In other words, a function ƒ is also a set ƒ where ƒ⊆X⨯Y

• Let x∈X, y∈Y, for a function ƒ from X to Y and (x,y)∈ƒ, if ƒ is a standard function, then the function ƒ can be expressed using the standard funcational notation y=f(x) in stead of using the relation notation xƒy.

• X is the domain set of ƒ and the range set of ƒ is {y∈Y:there exists an x∈X for which (x,y)∈ƒ}. Or imply for every x∈X, there is a y∈Y such that (x,y)∈ƒ.

• Since (x,y) is unique, if (x,y)∈ƒ and (x,z)∈ƒ, then y=z 

Function Operations

Although a function is usually defined by a definition, a function can also be produced by function operations.

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