October 19, 2017 15:00:03
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Math > Functions > Definitions

Definitions

The definitions of functions and related terms.

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Definition of a function [tags: function domain codomain]
A function from `A` to `B` is set of ordered pairs `S sube A xx B` such that:
  1. `(AA u in A)(EE v in B)((u,v) in S)`, and
  2. `(AA u in A)(AA v,w in B)((u,v) in S and (u,w) in S => v=w)`
For sets `A` and `B`, "`f:A->B`" means `f` is a function from `A` to `B`. If `f` is a function, then "`f(u)=v`" means `(u,v) in f`.

The set `A` is called the function's domain and the set `B` is called the function's codomain. The above definition basically says that a) for every element `u` in `A`, `f(u) in B`, and b) if `f(u)=v and f(u)=w`, then `v=w` (ie, each element of the function's domain maps to exactly one element of the function's codomain). The word "range" is sometimes used to refer to a function's codomain, and sometimes to refer to its image set.
Definition of image set [tags: imageSet]
The image set of a function is the subset of the function's codomain onto which elements of its domain are mapped.

The image set of a function `f:A->B`, is defined as
`{f(t):t in A}`
and may be written as `Im(f) or f(A)`.
Definition of surjective [tags: surjective]
A function is surjective if the function's image set equals its codomain.

A function `f:A->B` is surjective (or "onto") if
`(AA v in B)(EE u in A)(f(u)=v)
More concisely, `f` is surjective if `Im(f)=B`. This essentially means that every element of the function's codomain is mapped to by some element of its domain.
Definition of injective [tags: injective]
A function is injective (or "one-to-one") if no element of its codomain is mapped to by more than one element of its domain.

A function `f:A->B` is injective if
`(AA u,w in A)(f(u)=f(w) => u=w)`
This means that no element of the function's codomain is mapped to by more than one element of its domain. In order for a function to be invertible, it must be one-to-one.
Definition of bijective [tags: bijective]
A function is bijective if it is both surjective and injective.
Definition of composition [tags: composition]
Given `f:A->B` and `g:B->C`, the composition of `f` and `g` (written `g@f` and read "`g` composition `f`" or "`g` of `f`") is the subset of `A xx C` equal to
`{(u,w) in A xx C : g(f(u))=w}`
In other words, `(u,w) in g@f` means `w=g(f(u))`, and `(g@f)(x) = g(f(x))` for all `x`.
Definition of strictly increasing
For a function `f` with domain `RR`, `f` is strictly increasing if:
`(AA x,y in RR)(x<y => f(x)<f(y))`
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